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<div class="ChapSects"><a href="chap6_mj.html#X862C248A828A2C4A">6 <span class="Heading">Reidemeister zeta functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X862C248A828A2C4A">6.1 <span class="Heading">Reidemeister zeta functions</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X78F0CE5987B70AA2">6.1-1 ReidemeisterZetaCoefficients</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X79A2CD257BA1E037">6.1-2 IsRationalReidemeisterZeta</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7959DBAF78CC4401">6.1-3 ReidemeisterZeta</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X829058F97A8858F1">6.1-4 PrintReidemeisterZeta</a></span>
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</div>

<h3>6 <span class="Heading">Reidemeister zeta functions</span></h3>

<p><a id="X862C248A828A2C4A" name="X862C248A828A2C4A"></a></p>

<h4>6.1 <span class="Heading">Reidemeister zeta functions</span></h4>

<p>Let <span class="SimpleMath">\(\varphi,\psi\colon G \to G\)</span> be endomorphisms such that <span class="SimpleMath">\(R(\varphi^n,\psi^n) < \infty\)</span> for all <span class="SimpleMath">\(n \in \mathbb{N}\)</span>. Then the <em>Reidemeister zeta function</em> <span class="SimpleMath">\(Z_{\varphi,\psi}(s)\)</span> of the pair <span class="SimpleMath">\((\varphi,\psi)\)</span> is defined as</p>

<p class="center">\[Z_{\varphi,\psi}(s) := \exp \sum_{n=1}^\infty \frac{R(\varphi^n,\psi^n)}{n} s^n.\]</p>

<p>Please note that the functions below are only implemented for endomorphisms of finite groups.</p>

<p><a id="X78F0CE5987B70AA2" name="X78F0CE5987B70AA2"></a></p>

<h5>6.1-1 ReidemeisterZetaCoefficients</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReidemeisterZetaCoefficients</code>( <var class="Arg">endo1</var>[, <var class="Arg">endo2</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: two lists of integers.</p>

<p>For a finite group, the sequence of Reidemeister numbers of the iterates of <var class="Arg">endo1</var> and <var class="Arg">endo2</var>, i.e. the sequence <code class="code">R(<var class="Arg">endo1</var>,<var class="Arg">endo2</var>)</code>, <code class="code">R(<var class="Arg">endo1</var>^2,<var class="Arg">endo2</var>^2)</code>, ..., is eventually periodic. Thus there exist a periodic sequence <span class="SimpleMath">\((P_n)_{n \in \mathbb{N}}\)</span> and an eventually zero sequence <span class="SimpleMath">\((Q_n)_{n \in \mathbb{N}}\)</span> such that</p>

<p class="center">\[\forall n \in \mathbb{N}: R(\varphi^n,\psi^n) = P_n + Q_n.\]</p>

<p>This function returns two lists: the first list contains one period of the sequence <span class="SimpleMath">\((P_n)_{n \in \mathbb{N}}\)</span>, the second list contains <span class="SimpleMath">\((Q_n)_{n \in \mathbb{N}}\)</span> up to the part where it becomes the constant zero sequence.</p>

<p><a id="X79A2CD257BA1E037" name="X79A2CD257BA1E037"></a></p>

<h5>6.1-2 IsRationalReidemeisterZeta</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRationalReidemeisterZeta</code>( <var class="Arg">endo1</var>[, <var class="Arg">endo2</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> if the Reidemeister zeta function of <var class="Arg">endo1</var> and <var class="Arg">endo2</var> is rational, otherwise <code class="keyw">false</code>.</p>

<p><a id="X7959DBAF78CC4401" name="X7959DBAF78CC4401"></a></p>

<h5>6.1-3 ReidemeisterZeta</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReidemeisterZeta</code>( <var class="Arg">endo1</var>[, <var class="Arg">endo2</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the Reidemeister zeta function of <var class="Arg">endo1</var> and <var class="Arg">endo2</var> if it is rational, otherwise <code class="keyw">fail</code>.</p>

<p><a id="X829058F97A8858F1" name="X829058F97A8858F1"></a></p>

<h5>6.1-4 PrintReidemeisterZeta</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrintReidemeisterZeta</code>( <var class="Arg">endo1</var>[, <var class="Arg">endo2</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a string describing the Reidemeister zeta function of <var class="Arg">endo1</var> and <var class="Arg">endo2</var>.</p>

<p>This is often more readable than evaluating <code class="func">ReidemeisterZeta</code> (<a href="chap6_mj.html#X7959DBAF78CC4401"><span class="RefLink">6.1-3</span></a>) in an indeterminate, and does not require rationality.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">khi := GroupHomomorphismByImages( G, G, [ (1,2,3,4,5), (4,5,6) ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ (1,2,6,3,5), (1,4,5) ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ReidemeisterZetaCoefficients( khi );</span>
[ [ 7 ], [  ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRationalReidemeisterZeta( khi );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ReidemeisterZeta( khi );</span>
function( s ) ... end
<span class="GAPprompt">gap></span> <span class="GAPinput">s := Indeterminate( Rationals, "s" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ReidemeisterZeta( khi )(s);</span>
(1)/(-s^7+7*s^6-21*s^5+35*s^4-35*s^3+21*s^2-7*s+1)
<span class="GAPprompt">gap></span> <span class="GAPinput">PrintReidemeisterZeta( khi );</span>
"(1-s)^(-7)"
</pre></div>


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