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<div class="ChapSects"><a href="chap5_mj.html#X7B27E1F98083C837">5 <span class="Heading">Reidemeister numbers and spectra</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7FE8086286A91524">5.1 <span class="Heading">Reidemeister numbers</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8330E244852075A7">5.1-1 ReidemeisterNumber</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7CED57E379712C3A">5.2 <span class="Heading">Reidemeister spectra</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8777B3F77DBF01AF">5.2-1 ReidemeisterSpectrum</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8122B246860C1617">5.2-2 ExtendedReidemeisterSpectrum</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X78839C0886EBDB71">5.2-3 CoincidenceReidemeisterSpectrum</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7DB417F182B155C5">5.2-4 TotalReidemeisterSpectrum</a></span>
</div></div>
</div>

<h3>5 <span class="Heading">Reidemeister numbers and spectra</span></h3>

<p><a id="X7FE8086286A91524" name="X7FE8086286A91524"></a></p>

<h4>5.1 <span class="Heading">Reidemeister numbers</span></h4>

<p>The number of twisted conjugacy classes is called the Reidemeister number and is always a positive integer or infinity.</p>

<p><a id="X8330E244852075A7" name="X8330E244852075A7"></a></p>

<h5>5.1-1 ReidemeisterNumber</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReidemeisterNumber</code>( <var class="Arg">hom1</var>[, <var class="Arg">hom2</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">‣ NrTwistedConjugacyClasses</code>( <var class="Arg">hom1</var>[, <var class="Arg">hom2</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the Reidemeister number of ( <var class="Arg">hom1</var>, <var class="Arg">hom2</var> ).</p>

<p>If <span class="SimpleMath">\(G\)</span> is abelian, this function relies on (a generalisation of) <a href="chapBib_mj.html#biBjian83-a">[Jia83, Thm. 2.5]</a>. If <span class="SimpleMath">\(G = H\)</span>, <span class="SimpleMath">\(G\)</span> is finite non-abelian and <span class="SimpleMath">\(\psi = \operatorname{id}_G\)</span>, it relies on <a href="chapBib_mj.html#biBfh94-a">[FH94, Thm. 5]</a>. Otherwise, it simply calculates the twisted conjugacy classes and then counts them.</p>

<p><a id="X7CED57E379712C3A" name="X7CED57E379712C3A"></a></p>

<h4>5.2 <span class="Heading">Reidemeister spectra</span></h4>

<p>The set of all Reidemeister numbers of automorphisms is called the <em>Reidemeister spectrum</em> and is denoted by <span class="SimpleMath">\(\operatorname{Spec}_R(G)\)</span>, i.e.</p>

<p class="center">\[\operatorname{Spec}_R(G) := \{\, R(\varphi) \mid \varphi \in \operatorname{Aut}(G) \,\}.\]</p>

<p>The set of all Reidemeister numbers of endomorphisms is called the <em>extended Reidemeister spectrum</em> and is denoted by <span class="SimpleMath">\(\operatorname{ESpec}_R(G)\)</span>, i.e.</p>

<p class="center">\[\operatorname{ESpec}_R(G) := \{\, R(\varphi) \mid \varphi \in \operatorname{End}(G) \,\}.\]</p>

<p>The set of all Reidemeister numbers of pairs of homomorphisms from a group <span class="SimpleMath">\(H\)</span> to a group <span class="SimpleMath">\(G\)</span> is called the <em>coincidence Reidemeister spectrum</em> of <span class="SimpleMath">\(H\)</span> and <span class="SimpleMath">\(G\)</span> and is denoted by <span class="SimpleMath">\(\operatorname{CSpec}_R(H,G)\)</span>, i.e.</p>

<p class="center">\[\operatorname{CSpec}_R(H,G) := \{\, R(\varphi, \psi) \mid \varphi,\psi \in \operatorname{Hom}(H,G) \,\}.\]</p>

<p>If <var class="Arg">H</var> = <var class="Arg">G</var> this is also denoted by <span class="SimpleMath">\(\operatorname{CSpec}_R(G)\)</span>. The set of all Reidemeister numbers of pairs of homomorphisms from every group <span class="SimpleMath">\(H\)</span> to a group <span class="SimpleMath">\(G\)</span> is called the <em>total Reidemeister spectrum</em> and is denoted by <span class="SimpleMath">\(\operatorname{TSpec}_R(G)\)</span>, i.e.</p>

<p class="center">\[\operatorname{TSpec}_R(G) := \bigcup_{H} \operatorname{CSpec}_R(H,G).\]</p>

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

<p><a id="X8777B3F77DBF01AF" name="X8777B3F77DBF01AF"></a></p>

<h5>5.2-1 ReidemeisterSpectrum</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReidemeisterSpectrum</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the Reidemeister spectrum of <var class="Arg">G</var>.</p>

<p>If <span class="SimpleMath">\(G\)</span> is abelian, this function relies on the results from <a href="chapBib_mj.html#biBsend23-a">[Sen23]</a>. Otherwise, it relies on <a href="chapBib_mj.html#biBfh94-a">[FH94, Thm. 5]</a>.</p>

<p><a id="X8122B246860C1617" name="X8122B246860C1617"></a></p>

<h5>5.2-2 ExtendedReidemeisterSpectrum</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ExtendedReidemeisterSpectrum</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the extended Reidemeister spectrum of <var class="Arg">G</var>.</p>

<p>If <span class="SimpleMath">\(G\)</span> is abelian, this is just the set of all divisors of the order of <var class="Arg">G</var>. Otherwise, this function relies on <a href="chapBib_mj.html#biBfh94-a">[FH94, Thm. 5]</a>.</p>

<p><a id="X78839C0886EBDB71" name="X78839C0886EBDB71"></a></p>

<h5>5.2-3 CoincidenceReidemeisterSpectrum</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoincidenceReidemeisterSpectrum</code>( [<var class="Arg">H</var>, ]<var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the coincidence Reidemeister spectrum of <var class="Arg">H</var> and <var class="Arg">G</var>.</p>

<p><a id="X7DB417F182B155C5" name="X7DB417F182B155C5"></a></p>

<h5>5.2-4 TotalReidemeisterSpectrum</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TotalReidemeisterSpectrum</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the total Reidemeister spectrum of <var class="Arg">H</var> and <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Q := QuaternionGroup( 8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D := DihedralGroup( 8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ReidemeisterSpectrum( Q );</span>
[ 2, 3, 5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ExtendedReidemeisterSpectrum( Q );</span>
[ 1, 2, 3, 5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CoincidenceReidemeisterSpectrum( Q );</span>
[ 1, 2, 3, 4, 5, 8 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CoincidenceReidemeisterSpectrum( D, Q );</span>
[ 4, 8 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CoincidenceReidemeisterSpectrum( Q, D );</span>
[ 2, 3, 4, 6, 8 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">TotalReidemeisterSpectrum( Q );</span>
[ 1, 2, 3, 4, 5, 6, 8 ]
</pre></div>


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