Given a free group F of rank n, and a collection of reduced
 words w_1, ..., w_n in F, we wish to compute:

a. What is the rank of the smallest free factor of F containing these words.
b. Are these words "disk-busting"?  i.e. if H is a genus n handlebody,
 and if \alpha_i is a loop in H corresponding to w_i, then
 is there a compression disk for H which is disjoint from all the alpha_i's? 
 If not, then say the \alpha_i's (and w_i's) are disk-busting.


To get started:

Save file "whitehead.gap".
Start GAP Version 4 from same directory.
Type:
	  gap>Read("whitehead.gap");

Words in the free group are input as lists of numbers.
In free group of rank n, the numbers 1, ..., n correspond to
 free generators, and n+1, ..., 2n correspond to their inverses.

 For example, if F = <x_1, x_2> = free group of rank 2, then
 x_1*x_2 = [1,2]
 x_1*x_2^-1*x_1^-1 = [1,4,3]

Usage: whitehead(n, [ List_1, ... List_m ]);
where n = rank of F
List_1, ..., List_m are words in F.


Output is a 3-component List: [i, j, M], where

   i = 1 if the w_i's are disk-busting, i = 0 otherwise

   j = Rank of smallest free factor of F containing w_1, ..., w_m

   
     Let V = Abelianization of F = <x_1> + ... + <x_n>.
     Let D_1, ..., D_{n-j} be the maximal system of non-separating
     compressing disks for (H - \bigcup \alpha_i) found by Whitehead algorithm.
   
   M is a matrix, where row M[i] is the image of D[i] in V.


eg

	gap> Whitehead(3,[[1,2,3]]);         
	Graph reduced after 2 passes
	[ 0, 1, [ [ -1, 1, 0 ], [ 0, -1, 1 ] ] ]


So in free group of rank 3, x1*x_2*x_3 is not disk-busting;

is contained in a free subgroup of rank 1, and

there is a system of two non-separating compressing disks, whose images
     in H_1(F) are [-1,1,0] and [0,-1,1].


Also are told that the Whitehead algorithm required 2 Whitehead moves to
 reduce the Whitehead graph.

     

     gap> Whitehead(5, [[1,8,7],[6,6,3,3]]);
     Graph reduced after 2 passes
     [ 0, 3, [ [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ] ]


So in free group of rank 5,  the words x1*x3^-1*x7 and x1^-2*x3^2
 are not disk busting

they are contained in a free subgroup of rank 3, and

there is a system of two non-separating compressing disks, whose images
     in H_1(F) are [0,0,0,1,0] and [0,0,0,0,1].


	       gap> Whitehead(2, [[1,1,2,2]]);  
	       Graph reduced after 0 passes
	       [ 1, 2, [  ] ]

 
So in free group of rank 2, the element x_1^2*x_2^2 is disk-busting.
Whitehead graph is already reduced.