  
  [1X20 [33X[0;0YWords in free [22XZG[122X[101X[1X-modules[133X[101X
  
  
  [1X20.1 [33X[0;0Y [133X[101X
  
  [1X20.1-1 AddFreeWords[101X
  
  [33X[1;0Y[29X[2XAddFreeWords[102X( [3Xv[103X, [3Xw[103X ) [32X function[133X
  
  [33X[0;0YInputs  two  words [22Xv,w[122X in a free [22XZG[122X-module and returns their sum [22Xv+w[122X. If the
  characteristic  of  [22XZ[122X is greater than [22X0[122X then the next function might be more
  efficient.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X20.1-2 AddFreeWordsModP[101X
  
  [33X[1;0Y[29X[2XAddFreeWordsModP[102X( [3Xv[103X, [3Xw[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YInputs  two  words [22Xv,w[122X in a free [22XZG[122X-module and the characteristic [22Xp[122X of [22XZ[122X. It
  returns  the  sum  [22Xv+w[122X.  If  [22Xp=0[122X the previous function might be fractionally
  quicker.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X20.1-3 AlgebraicReduction[101X
  
  [33X[1;0Y[29X[2XAlgebraicReduction[102X( [3Xw[103X ) [32X function[133X
  [33X[1;0Y[29X[2XAlgebraicReduction[102X( [3Xw[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  word  [22Xw[122X  in a free [22XZG[122X-module and returns a reduced version of the
  word in which all pairs of mutually inverse letters have been cancelled. The
  reduction  is  performed in a free abelian group unless the characteristic [22Xp[122X
  of [22XZ[122X is entered.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X20.1-4 MultiplyWord[101X
  
  [33X[1;0Y[29X[2XMultiplyWord[102X( [3Xn[103X, [3Xw[103X ) [32X function[133X
  
  [33X[0;0YInputs a word [22Xw[122X and integer [22Xn[122X. It returns the scalar multiple [22Xn⋅ w[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X20.1-5 Negate[101X
  
  [33X[1;0Y[29X[2XNegate[102X( [[3Xi[103X, [3Xj[103X] ) [32X function[133X
  
  [33X[0;0YInputs a pair [22X[i,j][122X of integers and returns [22X[-i,j][122X.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X20.1-6 NegateWord[101X
  
  [33X[1;0Y[29X[2XNegateWord[102X( [3Xw[103X ) [32X function[133X
  
  [33X[0;0YInputs a word [22Xw[122X in a free [22XZG[122X-module and returns the negated word [22X-w[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X20.1-7 PrintZGword[101X
  
  [33X[1;0Y[29X[2XPrintZGword[102X( [3Xw[103X, [3Xelts[103X ) [32X function[133X
  
  [33X[0;0YInputs  a word [22Xw[122X in a free [22XZG[122X-module and a (possibly partial but sufficient)
  listing  elts  of  the  elements of [22XG[122X. The function prints the word [22Xw[122X to the
  screen in the form[133X
  
  [33X[0;0Y[22Xr_1E_1 + ... + r_nE_n[122X[133X
  
  [33X[0;0Ywhere  [22Xr_i[122X  are elements in the group ring [22XZG[122X, and [22XE_i[122X denotes the [22Xi[122X-th free
  generator of the module.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutPeriodic.html[107X) [133X
  
  [1X20.1-8 TietzeReduction[101X
  
  [33X[1;0Y[29X[2XTietzeReduction[102X( [3XS[103X, [3Xw[103X ) [32X function[133X
  
  [33X[0;0YInputs a set [22XS[122X of words in a free [22XZG[122X-module, and a word [22Xw[122X in the module. The
  function returns a word [22Xw'[122X such that {[22XS,w'[122X} generates the same abelian group
  as  {[22XS,w[122X}. The word [22Xw'[122X is possibly shorter (and certainly no longer) than [22Xw[122X.
  This function needs to be improved![133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X20.1-9 ResolutionBoundaryOfWord[101X
  
  [33X[1;0Y[29X[2XResolutionBoundaryOfWord[102X( [3XR[103X, [3Xn[103X, [3Xw[103X ) [32X function[133X
  
  [33X[0;0YInputs a resolution [22XR[122X, a positive integer [22Xn[122X and a list [22Xw[122X representing a word
  in  the  free  module [22XR_n[122X. It returns the image of [22Xw[122X under the [22Xn[122X-th boundary
  homomorphism.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
