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<p><a id="X8276B4377D092A80" name="X8276B4377D092A80"></a></p>
<div class="ChapSects"><a href="chap20_mj.html#X8276B4377D092A80">20 <span class="Heading"> Words in free <span class="SimpleMath">\(ZG\)</span>-modules </span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap20_mj.html#X7CFDEEC07F15CF82">20.1 <span class="Heading">  </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20_mj.html#X82D71B3F85D5BE77">20.1-1 AddFreeWords</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20_mj.html#X7CCBD61A7EEBD996">20.1-2 AddFreeWordsModP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20_mj.html#X7E3F96C27E64EEF9">20.1-3 AlgebraicReduction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20_mj.html#X844F2C187CC85C47">20.1-4 MultiplyWord</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20_mj.html#X7CA8A7CB81820EBB">20.1-5 Negate</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20_mj.html#X818F12EC81BA4788">20.1-6 NegateWord</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20_mj.html#X7FFAEBBD7B35F84C">20.1-7 PrintZGword</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20_mj.html#X7D769FB3873B9527">20.1-8 TietzeReduction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap20_mj.html#X7D043C4282B27B69">20.1-9 ResolutionBoundaryOfWord</a></span>
</div></div>
</div>

<h3>20 <span class="Heading"> Words in free <span class="SimpleMath">\(ZG\)</span>-modules </span></h3>

<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>

<h4>20.1 <span class="Heading">  </span></h4>

<p><a id="X82D71B3F85D5BE77" name="X82D71B3F85D5BE77"></a></p>

<h5>20.1-1 AddFreeWords</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AddFreeWords</code>( <var class="Arg">v</var>, <var class="Arg">w</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two words <span class="SimpleMath">\(v,w\)</span> in a free <span class="SimpleMath">\(ZG\)</span>-module and returns their sum <span class="SimpleMath">\(v+w\)</span>. If the characteristic of <span class="SimpleMath">\(Z\)</span> is greater than <span class="SimpleMath">\(0\)</span> then the next function might be more efficient.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7CCBD61A7EEBD996" name="X7CCBD61A7EEBD996"></a></p>

<h5>20.1-2 AddFreeWordsModP</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AddFreeWordsModP</code>( <var class="Arg">v</var>, <var class="Arg">w</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two words <span class="SimpleMath">\(v,w\)</span> in a free <span class="SimpleMath">\(ZG\)</span>-module and the characteristic <span class="SimpleMath">\(p\)</span> of <span class="SimpleMath">\(Z\)</span>. It returns the sum <span class="SimpleMath">\(v+w\)</span>. If <span class="SimpleMath">\(p=0\)</span> the previous function might be fractionally quicker.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7E3F96C27E64EEF9" name="X7E3F96C27E64EEF9"></a></p>

<h5>20.1-3 AlgebraicReduction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraicReduction</code>( <var class="Arg">w</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AlgebraicReduction</code>( <var class="Arg">w</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a word <span class="SimpleMath">\(w\)</span> in a free <span class="SimpleMath">\(ZG\)</span>-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 <span class="SimpleMath">\(p\)</span> of <span class="SimpleMath">\(Z\)</span> is entered.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X844F2C187CC85C47" name="X844F2C187CC85C47"></a></p>

<h5>20.1-4 MultiplyWord</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MultiplyWord</code>( <var class="Arg">n</var>, <var class="Arg">w</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a word <span class="SimpleMath">\(w\)</span> and integer <span class="SimpleMath">\(n\)</span>. It returns the scalar multiple <span class="SimpleMath">\(n\cdot w\)</span>.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7CA8A7CB81820EBB" name="X7CA8A7CB81820EBB"></a></p>

<h5>20.1-5 Negate</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Negate</code>( [<var class="Arg">i</var>, <var class="Arg">j</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a pair <span class="SimpleMath">\([i,j]\)</span> of integers and returns <span class="SimpleMath">\([-i,j]\)</span>.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X818F12EC81BA4788" name="X818F12EC81BA4788"></a></p>

<h5>20.1-6 NegateWord</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NegateWord</code>( <var class="Arg">w</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a word <span class="SimpleMath">\(w\)</span> in a free <span class="SimpleMath">\(ZG\)</span>-module and returns the negated word <span class="SimpleMath">\(-w\)</span>.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7FFAEBBD7B35F84C" name="X7FFAEBBD7B35F84C"></a></p>

<h5>20.1-7 PrintZGword</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrintZGword</code>( <var class="Arg">w</var>, <var class="Arg">elts</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a word <span class="SimpleMath">\(w\)</span> in a free <span class="SimpleMath">\(ZG\)</span>-module and a (possibly partial but sufficient) listing elts of the elements of <span class="SimpleMath">\(G\)</span>. The function prints the word <span class="SimpleMath">\(w\)</span> to the screen in the form</p>

<p><span class="SimpleMath">\(r_1E_1 + \ldots + r_nE_n\)</span></p>

<p>where <span class="SimpleMath">\(r_i\)</span> are elements in the group ring <span class="SimpleMath">\(ZG\)</span>, and <span class="SimpleMath">\(E_i\)</span> denotes the <span class="SimpleMath">\(i\)</span>-th free generator of the module.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutPeriodic.html">1</a></span> </p>

<p><a id="X7D769FB3873B9527" name="X7D769FB3873B9527"></a></p>

<h5>20.1-8 TietzeReduction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TietzeReduction</code>( <var class="Arg">S</var>, <var class="Arg">w</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a set <span class="SimpleMath">\(S\)</span> of words in a free <span class="SimpleMath">\(ZG\)</span>-module, and a word <span class="SimpleMath">\(w\)</span> in the module. The function returns a word <span class="SimpleMath">\(w'\) such that {\(S,w'\)</span>} generates the same abelian group as {<span class="SimpleMath">\(S,w\)</span>}. The word <span class="SimpleMath">\(w'\) is possibly shorter (and certainly no longer) than \(w\). This function needs to be improved!



<p><strong class="button">Examples:</strong></p>

<p><a id="X7D043C4282B27B69" name="X7D043C4282B27B69"></a></p>

<h5>20.1-9 ResolutionBoundaryOfWord</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionBoundaryOfWord</code>( <var class="Arg">R</var>, <var class="Arg">n</var>, <var class="Arg">w</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a resolution <span class="SimpleMath">\(R\)</span>, a positive integer <span class="SimpleMath">\(n\)</span> and a list <span class="SimpleMath">\(w\)</span> representing a word in the free module <span class="SimpleMath">\(R_n\)</span>. It returns the image of <span class="SimpleMath">\(w\)</span> under the <span class="SimpleMath">\(n\)</span>-th boundary homomorphism.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap14.html">1</a></span> </p>


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