Initial commit

Diagram/Diagram.java

@@ -0,0 +1,644 @@
+import javax.swing.*;
+import java.awt.*;
+import java.awt.event.*;
+import java.awt.geom.*;
+import java.net.*;
+import java.io.*;
+import java.applet.*;
+
+
+class Node {
+    //*********************************************
+    //*                                           *
+    //* This class is needed for the class list.  *
+    //* A Node looks like an element of the list. *
+    //*                                           *
+    //*********************************************
+    
+    private int value;      //Value of the Node 
+    private Node next;      //The next Node in the list
+    private Node previous;  //The previous Node in the list
+    
+    public Node(){
+	value=0;
+    }
+    
+    public Node(int _value){
+	value=_value;
+    }
+    
+    public void setValue(int _value){
+	value=_value;
+    }
+
+    public void setPrevious(Node _previous){
+	previous=_previous;
+    }
+    
+    public void setNext(Node _next){
+	next=_next;
+    }
+
+    public int getValue(){
+	return value;
+    }
+    
+    public Node getPrevious(){
+	return previous;
+    }
+    
+    public Node getNext(){
+	return next;
+    }
+}
+
+class List {
+    //******************************************
+    //*                                        *
+    //* This class decribe a bichained list.   *
+    //* I don't use the linkedList of java     *
+    //* because it need use of complex Object. *
+    //*                                        *
+    //******************************************
+
+    private Node first;   //The first Node of the list
+    private Node last;    //The last Node of the list
+    public Node current;  //A Node we play the game of an iterator
+    public int length;    //The length of the list
+    
+    public List(){
+	length=0;
+    }
+    
+    public Node getFirst(){
+	return first;
+    }	
+    
+    public Node getLast(){
+	return last;
+    }
+
+    public void initCurrent(){
+	current=first;
+    }
+    
+    public void setCurrent(Node _current){
+	current=_current;
+    }
+    
+    public Node getCurrent(){
+	return current;
+    }
+    
+    public int isEnd(){
+	if(current==last){
+	    return 1;
+	}
+	else{
+	    return 0;
+	}
+    }
+
+    public void shift(){
+	// ---------------------------------------
+	//| Change the current node with the next |
+	// ---------------------------------------
+	current=current.getNext();
+    }
+
+    public void addLast(int value){
+	// ----------------------------------------------------
+	//| Add a Node of value 'value' at the end of the list |
+	// ----------------------------------------------------
+	Node node=new Node(value);
+	if(length==0){
+	    first=node;
+	    last=node;
+	}
+	else{
+	    last.setNext(node);
+	    node.setPrevious(last);
+	    last=node;
+	}
+	length++;
+    }
+    
+    public Node addBefore(int value){
+	// -----------------------------------------------------------------
+	//| Add a Node of value 'value' before the current Node of the list | 
+	//| and return the new Node                                         |
+	// -----------------------------------------------------------------
+	Node node=new Node(value);
+	if(current==first){
+	    first=node;
+	    node.setNext(current);
+	    current.setPrevious(node);
+	}
+	else {
+	    Node temp=current.getPrevious();
+	    temp.setNext(node);
+	    node.setPrevious(temp);
+	    current.setPrevious(node);
+	    node.setNext(current);
+	}
+	length++;
+	return node;
+    }
+    
+    public Node addAfter(int value){
+	// ----------------------------------------------------------------
+	//| Add a Node of value 'value' after the current Node of the list | 
+	//| and return the new Node                                        |
+	// ----------------------------------------------------------------
+	Node node=new Node(value);
+	if(current==last){
+	    last=node;
+	    node.setPrevious(current);
+	    current.setNext(node);
+	}
+	else{
+	    Node temp=current.getNext();
+	    temp.setPrevious(node);
+	    node.setNext(temp);
+	    current.setNext(node);
+	    node.setPrevious(current);
+	}
+	length++;
+	return node;
+    }
+	       
+    public void remove(Node node){
+	// ----------------------------------
+	//| Remove the Node node of the list |
+	// ----------------------------------
+	if(length!=0){
+	    if(length==1){
+		length=0;
+	    }
+	    else{
+		if(node==first){
+		    first=node.getNext();
+		    node=first;
+		}
+		else if(node==last){
+		    last=node.getPrevious();
+		    current=last;
+		}
+		else{
+		    (node.getPrevious()).setNext(node.getNext());
+		    (node.getNext()).setPrevious(node.getPrevious());
+		}
+		length--;
+	    }
+	}
+    }
+    
+    public int value(){
+	return current.getValue();
+    }
+
+    public int length(){
+	return length;
+    }
+}	  
+ 
+public class Diagram extends JApplet implements ActionListener{
+    //*******************************************************
+    //*                                                     *
+    //* This class represent the window of the application. *
+    //*                                                     *
+    //*******************************************************
+
+
+    public static final long serialVersionUID = 0;
+    private Panel panel;           //Will draw graphics on the panel
+    private JTextField textField;  //An area in which will enter a braid word
+	    
+    public void init(){
+	JLabel label=new JLabel("Enter a word :");
+	label.setBounds(0,0,100,20);
+	add(label);
+	textField=new JTextField(10);
+	textField.addActionListener(this);  //Active the action listener of
+	textField.setBounds(100,0,380,20);
+	add(textField);
+	panel=new Panel();
+	panel.setBackground(Color.darkGray);
+	panel.setBounds(0,20,480,600);
+	add(panel);
+	add(new JLabel(""));
+    }
+
+    public void actionPerformed (ActionEvent event){
+	// ----------------------------------------------------------
+	//| Specify what the action on the element of the windows do |
+	// ----------------------------------------------------------
+	if(event.getSource()==textField){
+	    StringBuffer word=new StringBuffer(event.getActionCommand());
+	    panel.writeBraidWord(word);
+	}
+    }
+	
+    public String getAppletInfo() {
+        return "Handle reduction algorithm.";
+    }
+}
+    
+
+class Panel extends JPanel{
+    //************************************************
+    //*                                              *
+    //* This class is the key class of this program. *
+    //* It contains algoritms and drawing tools.     *
+    //*                                              *
+    //************************************************
+    
+    public static final long serialVersionUID = 0;
+
+    //=====> Braid variable
+    private StringBuffer braidWord;         //The initial braid word
+    private int braidWordSet=0;
+    private List braid;                     //The braid word as a list
+    
+    //=====> Strand variable
+    private int strandNumber=5;             //The number of strand
+    private float strandStep;               //The distance between two strand
+    private float strandWidth;              //The width of a strand
+    private float strandLeft;               //The x position of the leftmost strand
+    private float[] xPos;                   //The x postion of strands
+    private float[] xPosNext;               //The next x position of strands 
+    private float yUp;                      //The y position of the first element of the braid
+    
+    //======> Color variable
+    private Color[] strandColor;             //The color of different strand
+    private Color handleColor=Color.black;   //The color of the handle
+    private Color originalHandleStrandColor; //The original color of the handle strand
+      
+    //======> Way variable 
+    private GeneralPath[] continuousWay;     //The currant continuous way of strands
+    
+    //======> Double Buffering variable   
+    private Image offScreenBuffer;           //A buffer in wich we draw and after drawing we will show this buffer
+    private int isDraw;                      //For not repeat drawing
+
+    // =================
+    //I                 I
+    //I Braid functions I
+    //I                 I
+    // =================
+
+    private void initBraid(){
+	// -----------------------------------------------------
+	//| Iniliasition of braid as a list and of strandNumber |
+	// -----------------------------------------------------
+	int length;
+	int value;
+	char letter;
+	strandNumber=2;
+	value=0;
+	braid=new List();
+	length=braidWord.length();                 //get the length of the braid word 
+	braid.addLast(0);                          //add a trivial at the begin
+	for(int i=0;i<length;i++){
+	    letter=braidWord.charAt(i);
+	    if(letter>='a' && letter<='z'){        //if we have a lower case letter
+		value=(int)(letter-'a')+1;         //value is 1 for 'a', 2 for 'b', ...
+		if(value+1>strandNumber){          //update the strand number        
+		    strandNumber=value+1;     
+		}
+	    }
+	    else if(letter>='A' && letter<='Z'){   //if we have an upper case letter
+		value=-(int)(letter-'A')-1;        //value is -1 for 'A', -2 for 'B', ... 
+		if(-value+1>strandNumber){         //update the strand number
+		    strandNumber=-value+1;
+		}
+	    }
+	    else{
+		value=0;
+	    }
+	    braid.addLast(value);                  //add the value at the list of the braid
+	}
+	braid.addLast(0);                          //add a trivial at th end
+    }
+
+
+    public void writeBraidWord(StringBuffer buffer){
+	// -----------------------------------------------------
+	//| Initialise the braid word and some other parameters |
+	//------------------------------------------------------
+	braidWord=buffer;
+	initBraid();                              //initialise the braid list
+	isDraw=0;                                 //we draw a new image
+	braidWordSet=1;
+	update(super.getGraphics());              //update draw
+    }
+
+    
+    // =================
+    //I                 I
+    //I Color functions I
+    //I                 I
+    // =================
+    
+    public void initStrandColor(){
+	// ---------------------------
+	//| Init the color of strands |
+	// ---------------------------
+	strandColor=new Color[strandNumber];                         //Make the table of strand color
+	for(int strand=1;strand<strandNumber+1;strand++){
+	    if(strandNumber<10){
+		strandColor[strand-1]=Color.getHSBColor((float)1-(float)strand/(float)10,(float)0.8,1);     
+	    }
+	    else {
+		strandColor[strand-1]=Color.getHSBColor((float)1-(float)strand/(float)strandNumber,(float)0.8,1);     
+	    }	
+	}
+    }
+
+ 
+    // ==========================
+    //I                          I
+    //I Continuous way functions I
+    //I                          I
+    // ==========================
+
+    public void initContinuousWay(){
+	// -----------------------------------------
+	//| Start the continuous way for all strand |
+	// -----------------------------------------
+	continuousWay=new GeneralPath[strandNumber];             //Make the table of continuous way
+	xPos=new float[strandNumber];                            //Make the table of x position of strands
+	xPosNext=new float[strandNumber];                        //Make the table of next x position of strands
+	for(int strand=1;strand<strandNumber+1;strand++){
+	    continuousWay[strand-1]=new GeneralPath();           
+	    xPos[strand-1]=strandLeft+strand*strandStep;         //The xPos of n is strandLeft+n*strandStep
+	    continuousWay[strand-1].moveTo(xPos[strand-1],yUp);  //Start the continuous way at the y position yUp
+	}
+    }
+
+    public void closeContinuousWay(Graphics2D g,int strand){
+	// -------------------------------------------
+	//| Stop and draw the continous way of strand |
+	// -------------------------------------------
+	GeneralPath gp;
+	gp=continuousWay[strand-1];
+	g.setColor(strandColor[strand-1]);                                                            //Apply the color of strand
+	g.draw(gp);
+	gp=new GeneralPath();                                                                         //Make a new general path                                   
+	continuousWay[strand-1]=gp;                                                                   //Start the new continuous way
+    }
+   
+    public void permuteContinuousWay(int strand){
+	// ----------------------------------------------------------------- 
+	//| Echange the continuous way and the color of strand and strand+1 |
+	// -----------------------------------------------------------------
+	GeneralPath gp;
+	Color color;
+	//Echange continuous way 
+	gp=continuousWay[strand-1];
+	continuousWay[strand-1]=continuousWay[strand];
+	continuousWay[strand]=gp;
+	//Echange color
+	color=strandColor[strand-1];
+	strandColor[strand-1]=strandColor[strand];
+	strandColor[strand]=color;
+    }
+
+    // ================
+    //I                I
+    //I Math functions I
+    //I                I
+    // ================
+   
+    public double min(double a,double b){
+	if(a<b){
+	    return a;
+	}
+	else{
+	    return b;
+	}
+    }
+
+    public double max(double a,double b){
+	if(a>b){
+	    return a;
+	}
+	else{
+	    return b;
+	}
+    }
+
+    public double square(double a){
+	return a*a;
+    }
+
+    // ===================
+    //I                   I
+    //I Position function I
+    //I                   I
+    // ===================
+
+    public void calcXPosNext(){
+	// ----------------------------------------
+	//| Compute the nex x position of strands. |
+	//| It's a key function for animation.     |
+	// ----------------------------------------
+	for(int strand=0;strand<strandNumber;strand++){
+	    xPosNext[strand]=strandLeft+(strand+1)*strandStep;                                             //The general case
+	}
+    }
+
+    public void updateXPos(){
+	// ------------------------------------------------------------------
+	//| Update the x postion of strand with the nex x position of strand | 
+	// ------------------------------------------------------------------
+	for(int strand=0;strand<strandNumber;strand++){
+	    xPos[strand]=xPosNext[strand];
+	}	
+    }
+
+    public float[] calculIntersection(int strand,float yPos,int type){
+	// -----------------------------------------------------------------------
+	//| Compute the two key point for draw the generator strand |
+	// -----------------------------------------------------------------------
+	float []res=new float[4]; 
+	//res[0]<=res[1] is the x position of this points
+	//res[2]<=res[3] is the y position of this points
+	double decale=1.5*strandWidth;                                                      //Is distance of a strand with the other in the crossing
+	double distance;                         
+	if(type==1){                                                                        //If the crossing is positive 
+	    //We compute the distance between the left-up point and then right-down  
+	    distance=Math.sqrt(square(xPosNext[strand]-xPos[strand-1])+square(strandStep));
+	}
+	else {                                                                              //If the crossing is negative
+	    //We compute the distance between the right-up and the left-down 
+	    distance=Math.sqrt(square(xPos[strand]-xPosNext[strand-1])+square(strandStep));
+	}
+	double alpha=Math.acos(strandStep/distance);                                        //The angle of the crossing
+	double decalX=Math.sin(alpha)*decale;                                               //The x coords of decale
+	double decalY=Math.cos(alpha)*decale;                                               //The y coords of decale
+	
+	//The straight line D1 of equation x=m1*y+p1 contains the left-up and the right-down point
+	double m1=((double)strandStep)/((double)(xPosNext[strand]-xPos[strand-1]));
+	double p1=yPos-m1*((double)xPos[strand-1]);
+	//The straight line D2 of equation x=m2*y+p2 contains the right-up and the left-down point
+	double m2=((double)strandStep)/((double)(xPosNext[strand-1]-xPos[strand]));
+	double p2=yPos-m2*((double)xPos[strand]);
+	
+	double xInter=(p2-p1)/(m1-m2);                                                      //x coord of D1 inter D2               
+	double yInter=m1*xInter+p1;                                                         //y coord of D1 inter D2
+	
+	//Computaion of values of res, the min and max fonction are usefull for a good drawing
+	res[0]=(float)max(xInter-decalX,xPosNext[strand-1]);
+	res[1]=(float)min(xInter+decalX,xPosNext[strand]);
+	res[2]=(float)max(yInter-decalY,yPos);
+	res[3]=(float)min(yInter+decalY,yPos+strandStep);
+	return res;
+    }
+
+    // ===================
+    //I                   I
+    //I Drawing functions I
+    //I                   I
+    // ===================
+
+   
+    public void drawTrivial(float yPos){
+	// -------------------------------------
+	//| Draw a not extrem trivial generator |
+	// -------------------------------------
+	for(int strand=0;strand<strandNumber;strand++){
+	    continuousWay[strand].lineTo(xPosNext[strand],yPos+strandStep);
+	}
+    }
+    
+    public void drawNoCrossing(Graphics2D g,int strand,float yPos){
+	// ----------------------
+	//| Draw straight strand |
+	// ----------------------
+	continuousWay[strand-1].lineTo(xPosNext[strand-1],yPos+strandStep);
+    }
+
+    public void drawPositiveCrossing(Graphics2D g,int strand,float yPos){
+	// --------------------------
+	//| Draw a positive crossing |
+	// --------------------------
+	float inter[]=calculIntersection(strand,yPos,1);
+	continuousWay[strand-1].lineTo(inter[0],inter[2]);
+	closeContinuousWay(g,strand);	
+	continuousWay[strand-1].moveTo(inter[1],inter[3]);
+	continuousWay[strand-1].lineTo(xPosNext[strand],yPos+strandStep);
+	continuousWay[strand].lineTo(xPosNext[strand-1],yPos+strandStep);
+	permuteContinuousWay(strand);
+    }
+   
+    public void drawNegativeCrossing(Graphics2D g,int strand,float yPos){
+	// --------------------------
+	//| Draw a negative crossing |
+	// --------------------------
+	float inter[]=calculIntersection(strand,yPos,-1);
+	continuousWay[strand].lineTo(inter[1],inter[2]);
+	closeContinuousWay(g,strand+1);	
+	continuousWay[strand].moveTo(inter[0],inter[3]);
+	continuousWay[strand].lineTo(xPosNext[strand-1],yPos+strandStep);
+	continuousWay[strand-1].lineTo(xPosNext[strand],yPos+strandStep);
+	permuteContinuousWay(strand);
+    }
+
+    public void draw(Graphics g){
+	// --------------------------------------------
+	//| The main and first called drawing function |
+	// --------------------------------------------
+	Graphics2D g2=(Graphics2D) g;                                                            //A Graphics2D context is needed
+       	Rectangle r;                         
+	int generator;                                                                           //The absolute value of generator
+	int exponent;                                                                            //The sign of generator 1 or -1
+	float yPos;
+	//The current y position
+	r=getBounds();                                                                           //The dimension of the graphics window
+	g.setColor(Color.darkGray);                                                              //Background color       
+        g.clearRect(0,0,r.width,r.height);                                                       //Clear the drawing window 
+        g.drawRect(0,0,r.width-1,r.height-1);                                                    //Aply background color
+	
+	//Defintion of some strand parameter
+	strandStep=(float)min((double)r.width/(double)(strandNumber+1),(double)r.height/((double)(braid.length())));
+	strandWidth=strandStep/(float)5;
+	strandLeft=((float)r.width-((float)(strandNumber+1)*(float)strandStep+strandWidth))/(float)2;
+	yUp=(float)max((double)0,(double)((double)r.height-(double)strandStep*((double)(braid.length())))/(double)2);
+	
+	braid.initCurrent();                                                                     //Set current as braid.first  
+	initStrandColor();                                                                       //Init the strand color
+	initContinuousWay();                                                                     //Init then continuous way
+	g2.setStroke(new BasicStroke(strandWidth,BasicStroke.CAP_ROUND,BasicStroke.JOIN_BEVEL)); //Type of general path
+	yPos=yUp;                                                                                //Init current y position
+	int stop=0;                                                                              //stop is a flag for break the main while
+	while(stop==0){
+	    generator=braid.value();                                                             //Get braid value
+	    exponent=1;
+	    if(generator<0){                                                                     // \
+		exponent=-1;                                                                     //  Computation of generator and exponent
+		generator=-generator;                                                            // /
+	    }
+	    calcXPosNext();
+	    for(int strand=1;strand<strandNumber+1;strand++){
+		if(strand==generator && exponent==1){                                        //If the generator is positive        
+		    drawPositiveCrossing(g2,strand,yPos);
+		    strand++;
+		}
+		else if(strand==generator && exponent==-1){                                  //If the generato is negative 
+		    drawNegativeCrossing(g2,strand,yPos);
+		    strand++;
+		}
+		else{
+		    drawNoCrossing(g2,strand,yPos);
+		}
+	    }
+	    yPos+=strandStep;                                                                //Update yPos
+	    if(braid.isEnd()==1){                                                                //Are we at the end of the braid ?
+		stop=1;
+	    }
+	    else{
+		braid.shift();
+	    }
+	    updateXPos();                                                                        //Update xPos
+	}                                                                                        //Restart with the new generator
+	for(int strand=1;strand<strandNumber+1;strand++){                                         
+	    closeContinuousWay(g2,strand);                                                       //Close the continuous way
+	}
+    }
+
+    // ===================
+    //I                   I
+    //I Graphics function I
+    //I                   I
+    // ===================
+
+    public void paint(Graphics g){
+	if(isDraw==0){
+	    update(g);
+	    isDraw=1;
+	}
+	else{
+	    g.drawImage(offScreenBuffer,0,0,this);
+	}
+    }
+   
+    public void update(Graphics g) {
+	// -----------------------------------------------
+	//| Graphic update function with double-buffering |
+	// -----------------------------------------------
+	if(braidWordSet==1){
+	    Graphics gr; 
+	    if (offScreenBuffer==null||(!(offScreenBuffer.getWidth(this)==getBounds().width && offScreenBuffer.getHeight(this)==getBounds().height))){
+		offScreenBuffer=this.createImage(getBounds().width,getBounds().height);
+	    }
+	    gr=offScreenBuffer.getGraphics();
+	    draw(gr); 
+	    g.drawImage(offScreenBuffer,0,0,this);
+	}
+   }
+}
+
+
+
+

Diagram/diagram.html

@@ -0,0 +1,14 @@
+<html><head>
+<title>Braid applet, diagram page</title>   
+<script language="javascript"> </script> 
+</head>
+
+<body style="background-color: rgb(220, 220, 220);"> 
+<center><h1>Braid diagram applet</h1></center>
+
+<p><center>
+<applet code="Diagram.class" height="640" width="480">
+</applet>
+</p></center>
+
+</body>

HandleReduction.java

@@ -0,0 +1,1519 @@
+import javax.swing.*;
+import java.awt.*;
+import java.awt.event.*;
+import java.awt.geom.*;
+import java.net.*;
+import java.io.*;
+import java.applet.*;
+
+
+class Node {
+    //*********************************************
+    //*                                           *
+    //* This class is needed for the class list.  *
+    //* A Node looks like an element of the list. *
+    //*                                           *
+    //*********************************************
+    
+    private int value;      //Value of the Node 
+    private Node next;      //The next Node in the list
+    private Node previous;  //The previous Node in the list
+    
+    public Node(){
+	value=0;
+    }
+    
+    public Node(int _value){
+	value=_value;
+    }
+    
+    public void setValue(int _value){
+	value=_value;
+    }
+
+    public void setPrevious(Node _previous){
+	previous=_previous;
+    }
+    
+    public void setNext(Node _next){
+	next=_next;
+    }
+
+    public int getValue(){
+	return value;
+    }
+    
+    public Node getPrevious(){
+	return previous;
+    }
+    
+    public Node getNext(){
+	return next;
+    }
+}
+
+class List {
+    //******************************************
+    //*                                        *
+    //* This class decribe a bichained list.   *
+    //* I don't use the linkedList of java     *
+    //* because it need use of complex Object. *
+    //*                                        *
+    //******************************************
+
+    private Node first;   //The first Node of the list
+    private Node last;    //The last Node of the list
+    public Node current;  //A Node we play the game of an iterator
+    public int length;    //The length of the list
+    
+    public List(){
+	length=0;
+    }
+    
+    public Node getFirst(){
+	return first;
+    }	
+    
+    public Node getLast(){
+	return last;
+    }
+
+    public void initCurrent(){
+	current=first;
+    }
+    
+    public void setCurrent(Node _current){
+	current=_current;
+    }
+    
+    public Node getCurrent(){
+	return current;
+    }
+    
+    public boolean isEnd(){
+	return current==last;
+    }
+
+    public void shift(){
+	// ---------------------------------------
+	//| Change the current node with the next |
+	// ---------------------------------------
+	current=current.getNext();
+    }
+
+    public void addFirst(int value){
+	// ----------------------------------------------------
+	//| Add a Node of value 'value' at the end of the list |
+	// ----------------------------------------------------
+	Node node=new Node(value);
+	if(length==0){
+	    first=node;
+	    last=node;
+	}
+	else{
+	    first.setPrevious(node);
+	    node.setNext(first);
+	    first=node;
+	}
+	length++;
+    }
+
+    public void addLast(int value){
+	// ----------------------------------------------------
+	//| Add a Node of value 'value' at the end of the list |
+	// ----------------------------------------------------
+	Node node=new Node(value);
+	if(length==0){
+	    first=node;
+	    last=node;
+	}
+	else{
+	    last.setNext(node);
+	    node.setPrevious(last);
+	    last=node;
+	}
+	length++;
+    }
+    
+    public Node addBefore(int value){
+	// -----------------------------------------------------------------
+	//| Add a Node of value 'value' before the current Node of the list | 
+	//| and return the new Node                                         |
+	// -----------------------------------------------------------------
+	Node node=new Node(value);
+	if(current==first){
+	    first=node;
+	    node.setNext(current);
+	    current.setPrevious(node);
+	}
+	else {
+	    Node temp=current.getPrevious();
+	    temp.setNext(node);
+	    node.setPrevious(temp);
+	    current.setPrevious(node);
+	    node.setNext(current);
+	}
+	length++;
+	return node;
+    }
+    
+    public Node addAfter(int value){
+	// ----------------------------------------------------------------
+	//| Add a Node of value 'value' after the current Node of the list | 
+	//| and return the new Node                                        |
+	// ----------------------------------------------------------------
+	Node node=new Node(value);
+	if(current==last){
+	    last=node;
+	    node.setPrevious(current);
+	    current.setNext(node);
+	}
+	else{
+	    Node temp=current.getNext();
+	    temp.setPrevious(node);
+	    node.setNext(temp);
+	    current.setNext(node);
+	    node.setPrevious(current);
+	}
+	length++;
+	return node;
+    }
+	       
+    public void remove(Node node){
+	// ----------------------------------
+	//| Remove the Node node of the list |
+	// ----------------------------------
+	if(length!=0){
+	    if(length==1){
+		length=0;
+	    }
+	    else{
+		if(node==first){
+		    first=node.getNext();
+		    node=first;
+		}
+		else if(node==last){
+		    last=node.getPrevious();
+		    current=last;
+		}
+		else{
+		    (node.getPrevious()).setNext(node.getNext());
+		    (node.getNext()).setPrevious(node.getPrevious());
+		}
+		length--;
+	    }
+	}
+    }
+    
+    public int value(){
+	return current.getValue();
+    }
+
+    public int length(){
+	return length;
+    }
+}	  
+ 
+public class HandleReduction extends JApplet implements ActionListener{
+    //*******************************************************
+    //*                                                     *
+    //* This class represent the window of the application. *
+    //*                                                     *
+    //*******************************************************
+
+
+    public static final long serialVersionUID = 0;
+    private Panel panel;           //Will draw graphics on the panel
+    private JTextField textFieldDraw;
+    private JTextField textFieldReduce;
+    private JTextField textFieldCompare1;
+    private JTextField textFieldCompare2; 
+    private JButton draw; 
+    private JButton reduce;
+    private JButton compare;       
+    private JButton option;
+    private JButton startDraw;
+    private JButton startReduce;    
+    private JButton startCompare;
+    private JButton applyOption;
+    private JButton cancelDraw;
+    private JButton cancelReduce;
+    private JButton cancelCompare;
+    private JButton cancelOption;
+    private JFrame drawFrame;
+    private JFrame reduceFrame;
+    private JFrame compareFrame;
+    private JFrame optionFrame;
+    private JScrollBar delay;
+    private JLabel speed;
+    private JRadioButton continuous;
+    private JRadioButton stepbystep;
+    public JLabel currentWord;
+    public int speedValue=100;
+    public boolean continuousAnimation=true;
+    public int speedValue2;
+    public boolean continuousAnimation2;
+
+	    
+    public void init(){
+	panel=new Panel();
+	panel.setBackground(Color.darkGray);
+	panel.setBounds(0,20,480,600);
+	add(panel);
+	currentWord=new JLabel("No braid word");
+	currentWord.setBounds(0,0,480,20);
+	add(currentWord);
+	draw=new JButton("Draw");
+	draw.addActionListener(this);
+	draw.setBounds(0,620,120,20);
+	add(draw);
+	reduce=new JButton("Reduce");       
+	reduce.addActionListener(this);     
+	reduce.setBounds(120,620,120,20);
+	add(reduce);            
+	compare=new JButton("Compare");
+	compare.addActionListener(this);
+	compare.setBounds(240,620,120,20);
+	add(compare);
+	option=new JButton("Controls");
+	option.addActionListener(this);
+	option.setBounds(360,620,120,20);
+	add(option);
+	add(new JLabel(""));
+    }
+
+    public void start(){
+	update(getGraphics());
+	validate();
+    }
+	
+    public void lock(){
+	draw.setEnabled(false);
+	reduce.setEnabled(false);
+	compare.setEnabled(false);
+	option.setEnabled(false);
+    }
+
+    public void unlock(){
+	draw.setEnabled(true);
+	reduce.setEnabled(true);
+	compare.setEnabled(true);
+	option.setEnabled(true);
+    }
+
+
+    public void actionPerformed(ActionEvent event){
+	// ----------------------------------------------------------
+	//| Specify what the action on the element of the windows do |
+	// ----------------------------------------------------------
+
+	//===> Draw
+	If(event.getSource()==draw){
+	    drawFrame=new JFrame("Draw");
+	    drawFrame.setSize(300,105);
+	    drawFrame.setLocationRelativeTo(null);
+	    drawFrame.setDefaultCloseOperation(JFrame.DO_NOTHING_ON_CLOSE);	    
+	    drawFrame.setResizable(false);
+	    lock();	
+	    JLabel message=new JLabel("Enter a braid word : ");
+	    message.setBounds(5,5,290,20);
+	    textFieldDraw=new JTextField(10);
+	    textFieldDraw.setBounds(5,25,290,20);
+	    textFieldDraw.addActionListener(this);
+	    startDraw=new JButton("Draw");
+	    startDraw.addActionListener(this);
+	    startDraw.setBounds(20,55,120,20);
+	    cancelDraw=new JButton("Cancel");
+	    cancelDraw.addActionListener(this);
+	    cancelDraw.setBounds(160,55,120,20);
+	    drawFrame.add(message);
+	    drawFrame.add(textFieldDraw);
+	    drawFrame.add(startDraw);
+	    drawFrame.add(cancelDraw);
+	    drawFrame.add(new JLabel(""));
+	    drawFrame.setVisible(true);
+        }
+	else if(event.getSource()==startDraw){
+	    panel.init(this,speedValue,continuousAnimation);
+	    panel.writeBraidWord(textFieldDraw.getText());
+	    unlock();
+	    drawFrame.dispose();
+	}
+	else if(event.getSource()==cancelDraw){
+	    unlock();
+	    drawFrame.dispose();	    
+	}
+
+	//===> Reduce
+	else if(event.getSource()==reduce){
+	   	reduceFrame=new JFrame("Reduce");
+		reduceFrame.setSize(300,105);
+		reduceFrame.setLocationRelativeTo(null);
+		reduceFrame.setDefaultCloseOperation(JFrame.DO_NOTHING_ON_CLOSE);
+		reduceFrame.setResizable(false);	    
+		lock();
+		JLabel message=new JLabel("Enter a braid word : ");
+		message.setBounds(5,5,290,20);
+		textFieldReduce=new JTextField(10);
+		textFieldReduce.setBounds(5,25,290,20);
+		textFieldReduce.addActionListener(this);
+		startReduce=new JButton("Reduce");
+		startReduce.addActionListener(this);
+		startReduce.setBounds(20,55,120,20);
+		cancelReduce=new JButton("Cancel");
+		cancelReduce.addActionListener(this);
+		cancelReduce.setBounds(160,55,120,20);
+		reduceFrame.add(message);
+		reduceFrame.add(textFieldReduce);
+		reduceFrame.add(startReduce);
+		reduceFrame.add(cancelReduce);
+		reduceFrame.add(new JLabel(""));
+		reduceFrame.setVisible(true);
+	}
+	else if(event.getSource()==startReduce){
+	    textFieldReduce.setEnabled(false);
+	    if(!panel.working){
+		panel.init(this,speedValue,continuousAnimation);
+		panel.writeBraidWord(textFieldReduce.getText());
+		panel.working=true;
+		startReduce.setText("Continue");
+		startReduce.updateUI();
+	    }
+	    startReduce.setEnabled(false);
+	    boolean red=panel.reduce();
+	    String message;
+	    int ind=panel.indice();
+	    if(red){
+		if(ind==0){
+		    message="The final word is empty, so your initial braid word \n"+textFieldReduce.getText()+" is trivial.";
+		}
+		else{
+		    message="There is no more handle and the final word is nonempty, so your initial braid word\n "+textFieldReduce.getText()+" is not trivial.";
+		}
+		JOptionPane.showMessageDialog(null,message);
+		unlock();
+		panel.working=false;
+		reduceFrame.dispose();
+	    }
+	    else{
+		startReduce.setEnabled(true);
+	    }
+	}
+	else if(event.getSource()==cancelReduce){
+	    reduceFrame.dispose();
+	    unlock();
+	    panel.working=false;
+	}
+	
+	//===> Compare
+	else if(event.getSource()==compare){
+	    compareFrame=new JFrame("Compare");
+	    compareFrame.setSize(300,145);
+	    compareFrame.setLocationRelativeTo(null);
+	    compareFrame.setDefaultCloseOperation(JFrame.DO_NOTHING_ON_CLOSE);
+	    compareFrame.setResizable(false);
+	    lock();
+	    JLabel message1=new JLabel("Enter a first braid word : ");
+	    message1.setBounds(5,5,290,20);
+	    textFieldCompare1=new JTextField(10);
+	    textFieldCompare1.setBounds(5,25,290,20);
+	    JLabel message2=new JLabel("Enter a second braid word : ");
+	    message2.setBounds(5,45,290,20);
+	    textFieldCompare2=new JTextField(10);
+	    textFieldCompare2.setBounds(5,65,290,20);
+	    startCompare=new JButton("Compare");
+	    startCompare.addActionListener(this);
+	    startCompare.setBounds(20,95,120,20);
+	    cancelCompare=new JButton("Cancel");
+	    cancelCompare.addActionListener(this);
+	    cancelCompare.setBounds(160,95,120,20);
+	    compareFrame.add(message1);
+	    compareFrame.add(textFieldCompare1);
+	    compareFrame.add(message2);
+	    compareFrame.add(textFieldCompare2);
+	    compareFrame.add(startCompare);	    
+	    compareFrame.add(cancelCompare);
+	    compareFrame.add(new JLabel(""));
+	    compareFrame.setVisible(true);
+	}
+	else if(event.getSource()==startCompare){
+	    if(!panel.working){
+		String word1,word2;	    
+		word1=textFieldCompare1.getText();
+		word2=textFieldCompare2.getText();
+		textFieldCompare1.setEnabled(false);
+		textFieldCompare2.setEnabled(false);
+		panel.init(this,speedValue,continuousAnimation);
+		panel.compare(word1,word2);
+		startCompare.setText("Continue");
+		startCompare.updateUI();
+	    }
+	    boolean red;
+	    startCompare.setEnabled(false);
+	    red=panel.reduce();
+	    if(red){
+		String message;
+		int ind=panel.indice();
+		if(ind==0){
+		    message="The final word is empty,\nso your initial braid words "+textFieldCompare1.getText()+" and "+textFieldCompare2.getText()+" are equivalent.";
+		}
+		else if(ind>0){
+		    //smaller
+		    message="There is no more handle and the final word is nonempty,\nso your initial braid words "+textFieldCompare1.getText()+" and "+textFieldCompare2.getText()+" are not equivalent.";
+		}
+		else{
+		    //larger
+		    message="There is no more handle and the final word is nonempty,\nso your initial braid words "+textFieldCompare1.getText()+" and "+textFieldCompare2.getText()+" are not equivalent.";
+		}
+		JOptionPane.showMessageDialog(null,message);
+		compareFrame.dispose();
+		panel.working=false;
+		unlock();
+	    }
+	    else{
+		startCompare.setEnabled(true);
+	    }	
+	}
+	else if(event.getSource()==cancelCompare){
+	    compareFrame.dispose();
+	    unlock();
+	    panel.working=false;
+	}
+
+	//===> Option
+	else if(event.getSource()==option){
+	    optionFrame=new JFrame("Controls");
+	    optionFrame.setSize(300,110);
+	    optionFrame.setLocationRelativeTo(null);
+	    optionFrame.setDefaultCloseOperation(JFrame.DO_NOTHING_ON_CLOSE);
+	    optionFrame.setResizable(false);
+	    lock();
+	    continuous=new JRadioButton();
+	    stepbystep=new JRadioButton();
+	    ButtonGroup group = new ButtonGroup();
+	    continuous.setText("Continuous");
+	    stepbystep.setText("Step-by-step");
+	    continuous.addActionListener(this);
+	    stepbystep.addActionListener(this);
+	    continuousAnimation2=continuousAnimation;
+	    speedValue2=speedValue;
+	    if(continuousAnimation){
+		continuous.setSelected(true);
+	    }
+	    else{
+		stepbystep.setSelected(true);
+	    }
+	    group.add(continuous);
+	    group.add(stepbystep);
+	    JLabel label1=new JLabel("Mode :");
+	    delay=new JScrollBar(JScrollBar.HORIZONTAL,speedValue,0,0,100);
+	    delay.addAdjustmentListener(new java.awt.event.AdjustmentListener(){
+		    public void adjustmentValueChanged(AdjustmentEvent event){
+			scrollBarValueChanged(event);
+		    }
+		});
+	    speed=new JLabel("Speed : "+speedValue);
+  	    label1.setBounds(5,5,60,20);
+	    continuous.setBounds(65,5,100,20);
+	    stepbystep.setBounds(165,5,120,20);
+	    speed.setBounds(5,30,80,20);
+	    delay.setBounds(85,30,200,20);
+	    applyOption=new JButton("Apply");
+	    applyOption.addActionListener(this);
+	    applyOption.setBounds(20,60,120,20);
+	    cancelOption=new JButton("Cancel");
+	    cancelOption.addActionListener(this);
+	    cancelOption.setBounds(160,60,120,20);
+	    optionFrame.add(label1);
+	    optionFrame.add(continuous);
+	    optionFrame.add(stepbystep);
+	    optionFrame.add(speed);	  
+	    optionFrame.add(delay);
+	    optionFrame.add(applyOption);
+	    optionFrame.add(cancelOption);
+	    optionFrame.add(new JLabel(""));
+	    optionFrame.setVisible(true);
+	}
+	else if(event.getSource()==continuous){
+	    if(continuous.isSelected()){
+		continuousAnimation2=true;
+	    }
+	}
+	else if(event.getSource()==stepbystep){
+	    if(stepbystep.isSelected()){
+		continuousAnimation2=false;
+	    }
+	}
+	else if(event.getSource()==applyOption){
+	    continuousAnimation=continuousAnimation2;
+	    speedValue=speedValue2;
+	    unlock();
+	    optionFrame.dispose();
+	}
+	else if(event.getSource()==cancelOption){
+	    optionFrame.dispose();
+	    unlock();
+	}
+    }
+
+    public void scrollBarValueChanged(AdjustmentEvent event){
+	if(event.getSource()==delay){
+	    speedValue2=delay.getValue();
+	    speed.setText("Speed : "+speedValue2);
+	}
+    }
+    
+    public String getAppletInfo() {
+        return "Handle reduction algorithm.";
+    }
+}
+    
+
+class Pause extends Thread{
+    //*********************************
+    //*                               *
+    //* This class pause the program. *
+    //*                               *
+    //*********************************
+
+    public void pause(int delayValue){
+	try {
+	    sleep(delayValue);
+	}
+	catch (InterruptedException ex) {}
+    }
+}
+
+class Panel extends JPanel{
+    //************************************************
+    //*                                              *
+    //* This class is the key class of this program. *
+    //* It contains algoritms and drawing tools.     *
+    //*                                              *
+    //************************************************
+    
+    public static final long serialVersionUID = 0;
+
+    //Word variable
+    private HandleReduction myWindow;
+    private JLabel currentBraidWord;
+
+    //=====> Braid variable
+    private String braidWord;         //The initial braid word
+    private boolean braidWordSet=false;
+    private List braid;                     //The braid word as a list
+    public String currentWord="";
+    public boolean working=false;
+    private String compareWord1;
+    private String compareWord2;
+    
+
+    //=====> Strand variable
+    private int strandNumber=5;             //The number of strand
+    private float strandStep;               //The distance between two strand
+    private float strandWidth;              //The width of a strand
+    private float strandLeft;               //The x position of the leftmost strand
+    private float[] xPos;                   //The x postion of strands
+    private float[] xPosNext;               //The next x position of strands 
+    private float yUp;                      //The y position of the first element of the braid
+    
+    //=====> Handle variable
+    private Node handleBegin;                //The first node of the handle
+    private Node handleEnd;                  //The last node of the handle
+    private int handleStrand;                //The current strand of the handle
+    private int handleType;                  //The type of the handle, is 1 if the handle start with a positive element
+    private boolean handleNormalDraw;        //This variable is true if we draw the handle with the classic metod
+    private boolean onHandle;                //This variable is true if we are in between the handleBegin Node and the handleEnd node
+    private float handleBeginYPos;           //The y position of the begin of the handle
+    private float handleEndYPos;             //The y position of the end of the handle
+    
+    //======> Trivial variable
+    private int trivialNumber;               //The number of trivial element without the extremity
+    private float trivialHeight;             //The height of a trivial element
+    
+    //======> Animation variable
+    private int animationStep;               //The current animation step
+    private float parameterAnimationStep2;   // \
+    private float parameterAnimationStep4;   //  | Parameters for the animation 
+    private float parameterAnimationStep5;   // /
+    private boolean continuousAnimation;
+    private int delayValue; 
+    private Pause pause=new Pause();         //For make pause during animation   
+
+    //======> Color variable
+    private Color[] strandColor;             //The color of different strand
+    private Color handleColor=Color.black;   //The color of the handle
+    private Color originalHandleStrandColor; //The original color of the handle strand
+      
+    //======> Way variable 
+    private GeneralPath[] continuousWay;     //The currant continuous way of strands
+    
+    //======> Double Buffering variable   
+    private Image offScreenBuffer;           //A buffer in wich we draw and after drawing we will show this buffer
+    private boolean isDraw;                  //For not repeat drawing
+ 
+
+    // =================
+    //I                 I
+    //I Braid functions I
+    //I                 I
+    // =================
+
+    private void initBraid(){
+	// -----------------------------------------------------
+	//| Iniliasition of braid as a list and of strandNumber |
+	// -----------------------------------------------------
+	int length;
+	int value;
+	char letter;
+	strandNumber=2;
+	value=0;
+	braid=new List();
+	length=braidWord.length();                 //get the length of the braid word 
+	braid.addLast(0);                          //add a trivial at the begin
+	for(int i=0;i<length;i++){
+	    letter=braidWord.charAt(i);
+	    if(letter>='a' && letter<='z'){        //if we have a lower case letter
+		value=(int)(letter-'a')+1;         //value is 1 for 'a', 2 for 'b', ...
+		if(value+1>strandNumber){          //update the strand number        
+		    strandNumber=value+1;     
+		}
+	    }
+	    else if(letter>='A' && letter<='Z'){   //if we have an upper case letter
+		value=-(int)(letter-'A')-1;        //value is -1 for 'A', -2 for 'B', ... 
+		if(-value+1>strandNumber){         //update the strand number
+		    strandNumber=-value+1;
+		}
+	    }
+	    else{
+		value=0;
+	    }
+	    braid.addLast(value);                  //add the value at the list of the braid
+	}
+	braid.addLast(0);                          //add a trivial at th end
+    }
+    
+    public void init(HandleReduction _myWindow,int _delayValue,boolean _continuousAnimation){
+	myWindow=_myWindow;
+	delayValue=100-_delayValue;
+	continuousAnimation=_continuousAnimation;
+    }
+
+    public void writeBraidWord(String buffer){
+	// -----------------------------------------------------
+	//| Initialise the braid word and some other parameters |
+	//------------------------------------------------------
+	braidWord=buffer;
+	initBraid();                              //initialise the braid list
+	animationStep=0;                          //initialise the animation step
+	trivialNumber=0;                          // \
+	trivialHeight=0;                          //  Initilise the trivial parameters
+	handleNormalDraw=true;                    //we draw the handle with the classic method for the moment
+	updateCurrentBraidWord();
+	braidWordSet=true;
+	isDraw=false;                             //we draw a new image
+	update(super.getGraphics());              //update draw
+    }
+
+    public void compare(String word1, String word2){
+	int length;
+	int value;
+	char letter;
+	compareWord1=word1;
+	compareWord2=word2;
+	strandNumber=2;
+	value=0;
+	braid=new List();
+	length=word2.length();                 //get the length of the braid word          
+	for(int i=0;i<length;i++){
+	    letter=word2.charAt(i);
+	    if(letter>='a' && letter<='z'){        //if we have a lower case letter
+		value=(int)(letter-'a')+1;         //value is 1 for 'a', 2 for 'b', ...
+		if(value+1>strandNumber){          //update the strand number        
+		    strandNumber=value+1;     
+		}
+	    }
+	    else if(letter>='A' && letter<='Z'){   //if we have an upper case letter
+		value=-(int)(letter-'A')-1;        //value is -1 for 'A', -2 for 'B', ... 
+		if(-value+1>strandNumber){         //update the strand number
+		    strandNumber=-value+1;
+		}
+	    }
+	    else{
+		value=24;
+	    }
+	    braid.addLast(value);                  //add the value at the list of the braid
+	}
+	braid.addLast(0);
+	length=word1.length();                  
+	for(int i=0;i<length;i++){
+	    letter=word1.charAt(i);
+	    if(letter>='a' && letter<='z'){       
+		value=-(int)(letter-'a')-1;        
+		if(-value+1>strandNumber){        
+		    strandNumber=-value+1;     
+		}
+	    }
+	    else if(letter>='A' && letter<='Z'){ 
+		value=(int)(letter-'A')+1;         
+		if(value+1>strandNumber){       
+		    strandNumber=value+1;
+		}
+	    }
+	    else{
+		value=24;
+	    }
+	    braid.addFirst(value);                  
+	}
+	braid.addFirst(0);
+	animationStep=0;                          
+	trivialNumber=0;                          
+	trivialHeight=0;                         
+	handleNormalDraw=true;                      
+	updateCurrentBraidWord();
+	braidWordSet=true;
+	isDraw=false;                             //we draw a new image
+	working=true;
+	update(super.getGraphics());              //update draw
+    }                          
+
+    public void updateCurrentBraidWord(){
+	boolean stop;
+	int value;
+	String res="";
+	braid.initCurrent();
+	stop=false;
+	while(!stop){
+	    value=braid.value();
+	    if(value>0){
+		res+=(char)((value-1)+'a');
+	    }
+	    else if(value<0){
+		res+=(char)((-value-1)+'A');
+	    }
+	    if(braid.isEnd()){
+		stop=true;
+	    }
+	    if(!stop){
+		braid.shift();
+	    }
+	}
+	if(res==""){
+	    res="1";
+	}
+	currentWord=res;
+	(myWindow.currentWord).setText("Current braid word: "+res);
+	myWindow.update(myWindow.getGraphics());
+	myWindow.validate();
+    }
+
+
+    // ===============================
+    //I                               I
+    //I  Handle reduction algorithms  I
+    //I                               I
+    // ===============================
+
+    public void findHandle(){
+	// -----------------------------------
+	//| Search a handle in the braid word | 
+	//| with no intricated handle         |
+	// -----------------------------------
+	Node handles[]=new Node[strandNumber];     //We stock information of potential handle
+	int signs[]=new int[strandNumber];         //We stock the sign of the last n generato in signs[n];
+	int sign,value;
+	boolean stop;                                  //A flag for stop the main while
+	for(int i=0;i<strandNumber;i++){
+	    signs[i]=0;
+	}
+	stop=false;
+	value=0;
+	sign=0;
+	braid.initCurrent();                       //Initiliastion of braid current
+	while(!stop){
+	    value=braid.value();
+	    sign=1;
+	    if(value<0){                           //We compte the sign
+		value=-value;
+		sign=-1;
+	    }
+	    else if(value==0){                
+		sign=0;
+	    }
+	    if(signs[value]*sign<0){               //We have found the first handle with no imbricated handle
+		stop=true;
+	    }
+	    else{
+		signs[value]=sign;                 //Update signs
+		handles[value]=braid.getCurrent(); //Update value
+		for(int j=1;j<value;j++){          //Clear the potentialy imbricated handle
+		    signs[j]=0;
+		}
+	    }
+	    if(braid.isEnd()){                  //End of the braid ?
+		stop=true;
+	    }
+	    if(!stop){                           //Restart with the ext generator 
+		braid.shift(); 
+	    }
+	}
+	if(signs[value]*sign<0){                   //If we have found a handle
+	    handleBegin=handles[value];
+	    handleEnd=braid.getCurrent();
+	    handleStrand=value;
+	    handleType=-sign;
+	}
+	else{
+	    handleStrand=0;                        //If we haven't found handle
+	}
+    }
+
+    private void insertTrivials(){
+	// -----------------------------------------------------
+	//| This fonction insert trivials generato in the braid |
+        //| in order to make some space for the animation.      |
+	// -----------------------------------------------------
+	int value;
+	boolean stop;
+	stop=false;
+	braid.setCurrent(handleBegin);               //We start at the begin of the handle
+	trivialNumber=0;
+	while(!stop){
+	    value=braid.value();
+	    if(Math.abs(value)==handleStrand-1){     //If the generator is handleStrand-1 or -(handleStrand-1)
+		braid.addBefore(0);                  //We add a trivial before 
+		braid.setCurrent(braid.addAfter(0)); //We add a trivial after
+		trivialNumber+=2;                    //Update the number of trivials
+	    }
+	    braid.shift();
+	    if(braid.getCurrent()==handleEnd){       //If we are at the end of the handlee
+		stop=true;                           //we stop the main while
+	    }
+	}
+    }
+
+    public void removeHandle(){
+	// -------------------------------------------
+	//| This function remove the handle was found |
+	// -------------------------------------------
+	int sign;
+	int value;
+	boolean stop;
+	sign=-handleType;                                              //sign is a variable for alternating sign of new handleStrand generator
+	stop=false;
+	braid.setCurrent(handleBegin);                                 //We strat at the begin of the handle
+	while(!stop){
+	    value=braid.value();
+	    if(Math.abs(value)==handleStrand){                         //If the generator is handleStrand or -handleStrand
+		(braid.getCurrent()).setValue(0);                      //we replace it with a trivial generator
+	    }
+	    if(value==0){
+		(braid.getCurrent()).setValue((handleStrand-1)*sign);  //If the generator is a trivial we replace it
+		sign=-sign;                                            //with (handleStrand-1)*sign
+	    }
+	    if(value==handleStrand-1){                                 //If the genertaor is handleStrand-1
+		(braid.getCurrent()).setValue(value+1);                //we replace it with handleStrand
+	    }
+	    if(value==-(handleStrand-1)){                             //If the generator is -(handleStand-1)
+		(braid.getCurrent()).setValue(value-1);               //we replace it with -handleStrand
+	    }
+	    if(braid.getCurrent()==handleEnd){                        //If we are at he end of te endle
+		stop=true;                                               //we stop the main while
+	    }
+	    else{
+		braid.shift();                                        //else we restart with the next generator
+	    }
+	}
+	trivialNumber=2;                                              //At the end trivial numbers must be 2
+    }
+
+    private void removeTrivials(){
+	// -------------------------------------------
+	//| Remove the not extreme trivials generator | 
+	// -------------------------------------------
+	braid.remove(handleBegin);
+	braid.remove(handleEnd);
+	trivialNumber=0;
+    }
+
+    public int indice(){
+	int value;
+	int indice=0;
+	boolean stop=false;
+	braid.initCurrent();
+	while(!stop){
+	    value=braid.value();
+	    if(Math.abs(value)>Math.abs(indice)){
+		indice=value;
+	    }
+	    if(braid.isEnd()){
+		stop=true;  
+	    }
+	    else{        
+		braid.shift(); 
+	    }
+	}
+	return indice;
+    }
+    
+    // =================
+    //I                 I
+    //I Color functions I
+    //I                 I
+    // =================
+    
+    public void initStrandColor(){
+	// ---------------------------
+	//| Init the color of strands |
+	// ---------------------------
+	strandColor=new Color[strandNumber];                         //Make the table of strand color
+	for(int strand=1;strand<strandNumber+1;strand++){
+	    if(strandNumber<10){
+		strandColor[strand-1]=Color.getHSBColor((float)1-(float)strand/(float)10,(float)0.8,1);     
+	    }
+	    else {
+		strandColor[strand-1]=Color.getHSBColor((float)1-(float)strand/(float)strandNumber,(float)0.8,1);     
+	    }	
+	}
+    }
+
+    // ==================
+    //I                  I
+    //I Handle functions I
+    //I                  I
+    // ==================
+
+    public void colorizeHandle(Graphics2D g,float yPos){
+	// --------------------------------------
+	//| Colorize the handle if we are on her | 
+	// --------------------------------------
+	if(handleStrand!=0){                                                     //We verify we have found a handle
+	    if(braid.getCurrent()==handleBegin){                                 //If we are at the begin of the handle
+		handleBeginYPos=yPos;                                            //Initialize the y position of the begin of the handle
+		onHandle=true;                                                      //We are on the handle
+		closeContinuousWay(g,handleStrand);                              //Close the conitnuous way of handleStrand
+		originalHandleStrandColor=strandColor[handleStrand-1];           //Save the original color of the strand handleStrand
+		strandColor[handleStrand-1]=handleColor;                         //Change color of the strand handleStrand
+		continuousWay[handleStrand-1].moveTo(xPos[handleStrand-1],yPos); //Start position of the new continuous way of handleStrand
+	    }
+	    else if(braid.getCurrent()==handleEnd.getNext()){                    //If we are at the end of the handle
+		handleEndYPos=yPos;                                              //Initailaize the y position of the end of the handle
+		closeContinuousWay(g,handleStrand);                              //Close the continuous way of handleStrand
+		strandColor[handleStrand-1]=originalHandleStrandColor;           //The original color of strand handleStrand
+		continuousWay[handleStrand-1].moveTo(xPos[handleStrand-1],yPos); //Start position of the new continuous way
+		onHandle=false;                                                  //We are no longer in the handle
+	    }
+	}
+    }
+
+    public void changeHandleStrand(int generator){
+	// ------------------------------------------------------
+	//| Update the value of the strand on wich the handle is |
+	// ------------------------------------------------------
+	if(handleStrand!=0 && onHandle){
+	    if(generator==handleStrand){
+		handleStrand++;
+	    }
+	    else if(generator+1==handleStrand){
+		handleStrand--;
+	    }
+	}
+    }
+
+    // ==========================
+    //I                          I
+    //I Continuous way functions I
+    //I                          I
+    // ==========================
+
+    public void initContinuousWay(){
+	// -----------------------------------------
+	//| Start the continuous way for all strand |
+	// -----------------------------------------
+	continuousWay=new GeneralPath[strandNumber];             //Make the table of continuous way
+	xPos=new float[strandNumber];                            //Make the table of x position of strands
+	xPosNext=new float[strandNumber];                        //Make the table of next x position of strands
+	for(int strand=1;strand<strandNumber+1;strand++){
+	    continuousWay[strand-1]=new GeneralPath();           
+	    xPos[strand-1]=strandLeft+strand*strandStep;         //The xPos of n is strandLeft+n*strandStep
+	    continuousWay[strand-1].moveTo(xPos[strand-1],yUp);  //Start the continuous way at the y position yUp
+	}
+    }
+
+    public void closeContinuousWay(Graphics2D g,int strand){
+	// -------------------------------------------
+	//| Stop and draw the continous way of strand |
+	// -------------------------------------------
+	GeneralPath gp;
+	gp=continuousWay[strand-1];
+	g.setColor(strandColor[strand-1]);                                                            //Apply the color of strand
+	if(strand!=handleStrand || handleNormalDraw || strandColor[handleStrand-1]!=handleColor){ 
+	    //We verify we must draw the continuous way or not
+	    //We don't draw this way if we are on the handle and the value of handleNormalDraw is 0
+	    g.draw(gp);
+	}
+	gp=new GeneralPath();                                                                         //Make a new general path                                   
+	continuousWay[strand-1]=gp;                                                                   //Start the new continuous way
+    }
+   
+    public void permuteContinuousWay(int strand){
+	// ----------------------------------------------------------------- 
+	//| Echange the continuous way and the color of strand and strand+1 |
+	// -----------------------------------------------------------------
+	GeneralPath gp;
+	Color color;
+	//Echange continuous way 
+	gp=continuousWay[strand-1];
+	continuousWay[strand-1]=continuousWay[strand];
+	continuousWay[strand]=gp;
+	//Echange color
+	color=strandColor[strand-1];
+	strandColor[strand-1]=strandColor[strand];
+	strandColor[strand]=color;
+    }
+
+    // ================
+    //I                I
+    //I Math functions I
+    //I                I
+    // ================
+   
+    public double min(double a,double b){
+	if(a<b){
+	    return a;
+	}
+	else{
+	    return b;
+	}
+    }
+
+    public double max(double a,double b){
+	if(a>b){
+	    return a;
+	}
+	else{
+	    return b;
+	}
+    }
+
+    public double square(double a){
+	return a*a;
+    }
+
+    // ===================
+    //I                   I
+    //I Position function I
+    //I                   I
+    // ===================
+
+    public void calcXPosNext(){
+	// ----------------------------------------
+	//| Compute the nex x position of strands. |
+	//| It's a key function for animation.     |
+	// ----------------------------------------
+	for(int strand=0;strand<strandNumber;strand++){
+	    xPosNext[strand]=strandLeft+(strand+1)*strandStep;                                             //The general case
+	}
+	if(braid.getCurrent()==handleBegin){                                                               //If we have at the begin of the handle
+	    if(animationStep==2){                                                                          //If the step of animation is 2 
+		xPosNext[handleStrand]=(strandLeft+(handleStrand+parameterAnimationStep2)*strandStep);
+	    }
+	    else if(animationStep==4){                                                                     //If the step ofanimation is 4
+		xPosNext[handleStrand-1]+=parameterAnimationStep4*(float)strandStep;
+	    }
+	}
+	else if(braid.getCurrent()==handleEnd){                                                            //If we are at the end of the handle
+	}
+	else if(onHandle){                                                                              //If we are on the interior of the handle
+	    if(animationStep==2){                                                                          //If the step of animation is 2
+		xPosNext[handleStrand-1]=(strandLeft+(handleStrand+parameterAnimationStep2-1)*strandStep);
+	    }
+	    else if(animationStep==4){                                                                     //If the step of animation is 4
+		xPosNext[handleStrand-2]+=parameterAnimationStep4*(float)strandStep;
+		if(Math.abs(((braid.getCurrent()).getNext()).getValue())==handleStrand-2 ||
+		   Math.abs((braid.getCurrent()).getValue())==handleStrand-2){
+		    xPosNext[handleStrand-3]+=parameterAnimationStep4*(float)strandStep;
+		}
+	    }
+	    else if(animationStep==5){                                                                    //If the step of animation is 5
+		if(Math.abs((braid.getCurrent()).getValue())==handleStrand-1){		  
+		    xPosNext[handleStrand-2]+=parameterAnimationStep5*(float)strandStep;
+		}
+		else if(Math.abs((braid.getCurrent()).getValue())==handleStrand+1){
+		    xPosNext[handleStrand-1]+=parameterAnimationStep5*(float)strandStep;
+		}   
+	    }
+	}
+    }
+
+    public void updateXPos(){
+	// ------------------------------------------------------------------
+	//| Update the x postion of strand with the nex x position of strand | 
+	// ------------------------------------------------------------------
+	for(int strand=0;strand<strandNumber;strand++){
+	    xPos[strand]=xPosNext[strand];
+	}	
+    }
+    
+    public float[] calculIntersection(int strand,float yPos,int type){
+	// -----------------------------------------------------------------------
+	//| Compute the two key point for draw the generator strand |
+	// -----------------------------------------------------------------------
+	float []res=new float[4]; 
+	//res[0]<=res[1] is the x position of this points
+	//res[2]<=res[3] is the y position of this points
+	if(!handleNormalDraw && braid.getCurrent()==handleBegin){
+	    //If we are at the begin of handle and then handleNormal draw is 0 we send special values
+	    res[0]=xPosNext[strand-1];
+	    res[1]=xPosNext[strand-1];
+	    res[2]=yPos+strandStep;
+	    res[3]=yPos+strandStep;
+	    return res;
+	}
+	else if(!handleNormalDraw && braid.getCurrent()==handleEnd){
+	    //If we are at the end of handle and then handleNormal draw is 0 we send special value
+	    res[0]=xPosNext[strand];
+	    res[1]=xPosNext[strand];
+	    res[2]=yPos+strandStep;
+	    res[3]=yPos+strandStep;
+	    return res;
+	}
+	else{
+	    double decale=1.5*strandWidth;                                                      //Is distance of a strand with the other in the crossing
+	    double distance;                         
+	    if(type==1){                                                                        //If the crossing is positive 
+		//We compute the distance between the left-up point and then right-down  
+		distance=Math.sqrt(square(xPosNext[strand]-xPos[strand-1])+square(strandStep));
+	    }
+	    else {                                                                              //If the crossing is negative
+		//We compute the distance between the right-up and the left-down 
+		distance=Math.sqrt(square(xPos[strand]-xPosNext[strand-1])+square(strandStep));
+	    }
+	    double alpha=Math.acos(strandStep/distance);                                        //The angle of the crossing
+	    double decalX=Math.sin(alpha)*decale;                                               //The x coords of decale
+	    double decalY=Math.cos(alpha)*decale;                                               //The y coords of decale
+	    
+	    //The straight line D1 of equation x=m1*y+p1 contains the left-up and the right-down point
+	    double m1=((double)strandStep)/((double)(xPosNext[strand]-xPos[strand-1]));
+	    double p1=yPos-m1*((double)xPos[strand-1]);
+	    //The straight line D2 of equation x=m2*y+p2 contains the right-up and the left-down point
+	    double m2=((double)strandStep)/((double)(xPosNext[strand-1]-xPos[strand]));
+	    double p2=yPos-m2*((double)xPos[strand]);
+	    
+	    double xInter=(p2-p1)/(m1-m2);                                                      //x coord of D1 inter D2               
+	    double yInter=m1*xInter+p1;                                                         //y coord of D1 inter D2
+	
+	    //Computaion of values of res, the min and max fonction are usefull for a good drawing
+	    res[0]=(float)max(xInter-decalX,xPosNext[strand-1]);
+	    res[1]=(float)min(xInter+decalX,xPosNext[strand]);
+	    res[2]=(float)max(yInter-decalY,yPos);
+	    res[3]=(float)min(yInter+decalY,yPos+strandStep);
+	    return res;
+	}
+    }
+
+    // ===================
+    //I                   I
+    //I Drawing functions I
+    //I                   I
+    // ===================
+
+    public void drawHandle(Graphics2D g){
+	// -------------------------------------------
+	//| Draw the handle if handleNormal draw is 0 |
+	// -------------------------------------------
+	GeneralPath gp=new GeneralPath();
+	gp.moveTo(strandLeft+handleStrand*strandStep,handleBeginYPos); 
+	gp.lineTo(strandLeft+handleStrand*strandStep,handleEndYPos);
+	g.setColor(handleColor);                                         //Apply the color of the handle
+	g.draw(gp);
+    }
+    
+    public void drawTrivial(float yPos){
+	// -------------------------------------
+	//| Draw a not extrem trivial generator |
+	// -------------------------------------
+	for(int strand=0;strand<strandNumber;strand++){
+	    continuousWay[strand].lineTo(xPosNext[strand],yPos+trivialHeight*strandStep);
+	}
+    }
+    
+    public void drawNoCrossing(Graphics2D g,int strand,float yPos){
+	// ----------------------
+	//| Draw straight strand |
+	// ----------------------
+	continuousWay[strand-1].lineTo(xPosNext[strand-1],yPos+strandStep);
+    }
+
+    public void drawPositiveCrossing(Graphics2D g,int strand,float yPos){
+	// --------------------------
+	//| Draw a positive crossing |
+	// --------------------------
+	float inter[]=calculIntersection(strand,yPos,1);
+	continuousWay[strand-1].lineTo(inter[0],inter[2]);
+	closeContinuousWay(g,strand);	
+	continuousWay[strand-1].moveTo(inter[1],inter[3]);
+	continuousWay[strand-1].lineTo(xPosNext[strand],yPos+strandStep);
+	continuousWay[strand].lineTo(xPosNext[strand-1],yPos+strandStep);
+	permuteContinuousWay(strand);
+    }
+   
+    public void drawNegativeCrossing(Graphics2D g,int strand,float yPos){
+	// --------------------------
+	//| Draw a negative crossing |
+	// --------------------------
+	float inter[]=calculIntersection(strand,yPos,-1);
+	continuousWay[strand].lineTo(inter[1],inter[2]);
+	closeContinuousWay(g,strand+1);	
+	continuousWay[strand].moveTo(inter[0],inter[3]);
+	continuousWay[strand].lineTo(xPosNext[strand-1],yPos+strandStep);
+	continuousWay[strand-1].lineTo(xPosNext[strand],yPos+strandStep);
+	permuteContinuousWay(strand);
+    }
+
+    public void draw(Graphics g){
+	// --------------------------------------------
+	//| The main and first called drawing function |
+	// --------------------------------------------
+	Graphics2D g2=(Graphics2D) g;                                                            //A Graphics2D context is needed
+       	Rectangle r;                         
+	int generator;                                                                           //The absolute value of generator
+	int exponent;                                                                            //The sign of generator 1 or -1
+	float yPos;                                                                              //The current y position
+	r=getBounds();                                                                           //The dimension of the graphics window
+	g.setColor(Color.darkGray);                                                              //Background color       
+        g.clearRect(0,0,r.width,r.height);                                                       //Clear the drawing window 
+        g.drawRect(0,0,r.width-1,r.height-1);                                                    //Aply background color
+	
+	//Defintion of some strand parameter
+	strandStep=(float)min((double)r.width/(double)(strandNumber+1),(double)r.height/((double)(braid.length())-((double)1-(double)trivialHeight)*(double)trivialNumber));
+	strandWidth=strandStep/(float)5;
+	strandLeft=((float)r.width-((float)(strandNumber+1)*(float)strandStep+strandWidth))/(float)2;
+	yUp=(float)max((double)0,(double)((double)r.height-(double)strandStep*((double)(braid.length())-((double)1-(double)trivialHeight)*(double)trivialNumber))/(double)2);
+	
+	braid.initCurrent();                                                                     //Set current as braid.first  
+	initStrandColor();                                                                       //Init the strand color
+	initContinuousWay();                                                                     //Init then continuous way
+	g2.setStroke(new BasicStroke(strandWidth,BasicStroke.CAP_ROUND,BasicStroke.JOIN_BEVEL)); //Type of general path
+	yPos=yUp;                                                                                //Init current y position
+	int stop=0;                                                                              //stop is a flag for break the main while
+	if(!handleNormalDraw && handleType==1){
+	    //If handleNormalDraw is 0 and hande type is 1 we draw the handle first
+	    drawHandle(g2);
+	}
+	while(stop==0){
+	    generator=braid.value();                                                             //Get braid value
+	    exponent=1;
+	    if(generator<0){                                                                     // \
+		exponent=-1;                                                                     //  Computation of generato and exponent
+		generator=-generator;                                                            // /
+	    }
+	    colorizeHandle(g2,yPos);
+	    calcXPosNext();
+	    if(onHandle && generator==0){                                                     //If the generator is trivial and we are on the handle
+		drawTrivial(yPos);
+		yPos+=trivialHeight*strandStep;                                                  //Update yPos
+	    }
+	    else{
+		for(int strand=1;strand<strandNumber+1;strand++){
+		    if(strand==generator && exponent==1){                                        //If the generator is positive        
+			drawPositiveCrossing(g2,strand,yPos);
+			strand++;
+		    }
+		    else if(strand==generator && exponent==-1){                                  //If the generato is negative 
+			drawNegativeCrossing(g2,strand,yPos);
+			strand++;
+		    }
+		    else{
+			drawNoCrossing(g2,strand,yPos);
+		    }
+		}
+		changeHandleStrand(generator);                                                   //Update handle strand
+		yPos+=strandStep;                                                                //Update yPos
+	    }
+	    if(braid.isEnd()){                                                                //Are we at the end of the braid ?
+		stop=1;
+	    }
+	    else{
+		braid.shift();
+	    }
+	    updateXPos();                                                                        //Update xPos
+	}                                                                                        //Restart with the new generator
+	for(int strand=1;strand<strandNumber+1;strand++){                                         
+	    closeContinuousWay(g2,strand);                                                       //Close the continuous way
+	}
+	if(!handleNormalDraw && handleType==-1){
+	    //If handleNormalDraw is 0 and hande type is 1 we draw the handle last
+	    drawHandle(g2);
+	}
+    }
+
+    // ===================
+    //I                   I
+    //I Graphics function I
+    //I                   I
+    // ===================
+
+    public void paint(Graphics g){
+	if(!isDraw){
+	    update(g);
+	    isDraw=true;
+	}
+	else{
+	    g.drawImage(offScreenBuffer,0,0,this);
+	}
+    }
+   
+    public void update(Graphics g) {
+	// -----------------------------------------------
+	//| Graphic update function with double-buffering |
+	// -----------------------------------------------
+	if(braidWordSet){
+	    Graphics gr; 
+	    if (offScreenBuffer==null||(!(offScreenBuffer.getWidth(this)==getBounds().width && offScreenBuffer.getHeight(this)==getBounds().height))){
+		offScreenBuffer=this.createImage(getBounds().width,getBounds().height);
+	    }
+	    gr=offScreenBuffer.getGraphics();
+	    draw(gr); 
+	    g.drawImage(offScreenBuffer,0,0,this);
+	}
+   }
+
+    // ====================
+    //I                    I
+    //I Animation function I
+    //I                    I
+    // ====================
+
+    public boolean reduce(){
+	if(!braidWordSet){
+	    JOptionPane.showMessageDialog(null,"You must enter a braid word first ! ","Error",JOptionPane.ERROR_MESSAGE);
+	    return true;
+	}
+	else if(animationStep==0){
+	    findHandle();
+	    update(getGraphics());
+	    if(handleStrand==0){
+		//update(getGraphics());
+		return true;
+	    }
+	    else{
+		update(getGraphics());
+		insertTrivials();
+		trivialHeight=0;
+		animationStep=1;
+		handleNormalDraw=true;
+		if(continuousAnimation){
+		    return reduce();
+		}
+		else{
+		    return false;
+		}
+	    }
+	}
+	else if(animationStep==1){
+	    if(trivialHeight>=1){
+		trivialHeight=1;
+		parameterAnimationStep2=1;	    
+		animationStep=2;
+		pause.pause(delayValue);
+		return reduce();
+	    }
+	    else{
+		trivialHeight+=0.1;
+		update(getGraphics());
+		pause.pause(delayValue);
+		return reduce();
+	    }
+	}
+	else if(animationStep==2){
+	    if(handleStrand==0){
+		return true;
+	    }
+	    else if(parameterAnimationStep2<(float)(1.25)*((float)strandWidth/strandStep)){
+		handleNormalDraw=false;
+		animationStep=3;
+		pause.pause(delayValue);
+		return reduce();
+	    }
+	    else{
+		parameterAnimationStep2-=(float)0.05;
+		update(getGraphics());	
+		pause.pause(delayValue);
+		return reduce();
+	    }
+	}
+	else if(animationStep==3){
+	    parameterAnimationStep4=0;
+	    animationStep=4;
+	    update(getGraphics());
+	    pause.pause(delayValue);
+	    return reduce();
+	}
+	else if(animationStep==4){
+	    if(parameterAnimationStep4>=1){
+		removeHandle();
+		handleNormalDraw=true;
+		parameterAnimationStep5=1-(float)(1.25)*((float)strandWidth/strandStep);
+		animationStep=5;
+		pause.pause(delayValue);
+		return reduce();
+	    }
+	    else{
+		parameterAnimationStep4+=(float)0.05;
+		update(getGraphics());	
+		pause.pause(delayValue);
+		return reduce();
+	    }
+	}
+	else if(animationStep==5){
+	    if(parameterAnimationStep5<=(float)0.05){
+		parameterAnimationStep5=0;
+		animationStep=6;
+		pause.pause(delayValue);
+		return reduce();
+	    }
+	    else{
+		parameterAnimationStep5-=(float)0.05;
+		update(getGraphics());	
+		pause.pause(delayValue);
+		return reduce();
+	    }
+	}
+	else if(animationStep==6){
+	    if(trivialHeight<=0){
+		trivialHeight=0;
+		animationStep=7;
+		pause.pause(delayValue);
+		update(getGraphics());
+		return reduce();
+	    }
+	    else{
+		trivialHeight-=0.1;
+		update(getGraphics());	
+		pause.pause(delayValue);
+		return reduce();
+	    }
+	}
+	else if(animationStep==7){
+	    removeTrivials();
+	    handleStrand=0;
+	    update(getGraphics());	
+	    animationStep=0;
+	    updateCurrentBraidWord();
+	    update(getGraphics());	
+	    pause.pause(delayValue);
+	    if(continuousAnimation){
+		return reduce();
+	    }
+	    else{
+		return false;
+	    }
+	}
+	return false;
+    }
+}
+
+
+
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