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grahamknockillaree authored Jan 16, 2024
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31 changes: 19 additions & 12 deletions tutorial/chap5.html
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5.html#X86FD0A867EC9E64F">5.5-1 <span class="Heading">Non-trivial cup product</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5.html#X802C7755851A2530">5.5-2 <span class="Heading">Explicit first homology generators</span></a>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5.html#X783EF0F17B629C46">5.5-2 <span class="Heading">Explicit homology generators</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap5.html#X80D0D8EB7BCD05E9">5.6 <span class="Heading">Knotted proteins</span></a>
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</pre></div>

<p><a id="X802C7755851A2530" name="X802C7755851A2530"></a></p>
<p><a id="X783EF0F17B629C46" name="X783EF0F17B629C46"></a></p>

<h5>5.5-2 <span class="Heading">Explicit first homology generators</span></h5>
<h5>5.5-2 <span class="Heading">Explicit homology generators</span></h5>

<p>It could be desirable to obtain explicit representatives of the two homology generators in degree <span class="SimpleMath">1</span> that "persist". The following commands visualize the two explicit homology generators by visualizing a subspace <span class="SimpleMath">D⊂ X_10</span> where <span class="SimpleMath">D</span> is homotopy equivalent to a wedge of two circles and where the inclusion induces an isomorphism on first homology groups.</p>
<p>It could be desirable to obtain explicit representatives of the persistent homology generators that "persist" through a significant sequence of filtration terms. There are two such generators in degree <span class="SimpleMath">1</span> and one such generator in degree <span class="SimpleMath">2</span>. The explicit representatives in degree <span class="SimpleMath">n</span> could consist of an inclusion of pure cubical complexes <span class="SimpleMath">Y_n ⊂ X_10</span> for which the incuced homology homomorphism <span class="SimpleMath">H_n(Y_n, Z) → H_n(X_10, Z)</span> is an isomorphism, and for which <span class="SimpleMath">Y_n</span> is minimal in the sense that its homotopy type changes if any one or more of its top dimensional cells are removed. Ideally the space <span class="SimpleMath">Y_n</span> should be as "close to the original dataset" <span class="SimpleMath">X_0</span>. The following commands first construct an explicit degree <span class="SimpleMath">2</span> homology generator representative <span class="SimpleMath">Y_2⊂ X_10</span> where <span class="SimpleMath">Y_2</span> is homotopy equivalent to <span class="SimpleMath">X_10</span>. They then construct an explicit degree <span class="SimpleMath">1</span> homology generators representative <span class="SimpleMath">Y_1⊂ X_10</span> where <span class="SimpleMath">Y_1</span> is homotopy equivalent to a wedge of two circles. The final command displays the homology generators representative <span class="SimpleMath">Y_1</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Y:=FiltrationTerm(F,10);; </span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">for n in Reversed([1..9]) do</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">Y:=ContractedComplex(Y,FiltrationTerm(F,n));</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Y2:=FiltrationTerm(F,10);; </span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">for t in Reversed([1..9]) do</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">Y2:=ContractedComplex(Y2,FiltrationTerm(F,t));</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">od;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Y:=ContractedComplex(Y);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">R:=PureComplexRandomCell(Y);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">D:=PureComplexDifference(Y,R);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">D:=ContractedComplex(D);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Display(D);</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Y2:=ContractedComplex(Y2);;</span>

<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(FiltrationTerm(F,10));</span>
918881
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(Y2); </span>
61618

<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Y1:=PureComplexDifference(Y2,PureComplexRandomCell(Y2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Y1:=ContractedComplex(Y1);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(Y1);</span>
474
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Display(Y1);</span>

</pre></div>

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46 changes: 31 additions & 15 deletions tutorial/chap5.txt
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5.5-2 Explicit first homology generators

It could be desirable to obtain explicit representatives of the two homology
generators in degree 1 that "persist". The following commands visualize the
two explicit homology generators by visualizing a subspace D⊂ X_10 where D
is homotopy equivalent to a wedge of two circles and where the inclusion
induces an isomorphism on first homology groups.
5.5-2 Explicit homology generators

It could be desirable to obtain explicit representatives of the persistent
homology generators that "persist" through a significant sequence of
filtration terms. There are two such generators in degree 1 and one such
generator in degree 2. The explicit representatives in degree n could
consist of an inclusion of pure cubical complexes Y_n ⊂ X_10 for which the
incuced homology homomorphism H_n(Y_n, Z) → H_n(X_10, Z) is an isomorphism,
and for which Y_n is minimal in the sense that its homotopy type changes if
any one or more of its top dimensional cells are removed. Ideally the space
Y_n should be as "close to the original dataset" X_0. The following commands
first construct an explicit degree 2 homology generator representative Y_2⊂
X_10 where Y_2 is homotopy equivalent to X_10. They then construct an
explicit degree 1 homology generators representative Y_1⊂ X_10 where Y_1 is
homotopy equivalent to a wedge of two circles. The final command displays
the homology generators representative Y_1.

 Example 
gap> Y:=FiltrationTerm(F,10);; 
gap> for n in Reversed([1..9]) do
> Y:=ContractedComplex(Y,FiltrationTerm(F,n));
gap> Y2:=FiltrationTerm(F,10);; 
gap> for t in Reversed([1..9]) do
> Y2:=ContractedComplex(Y2,FiltrationTerm(F,t));
> od;
gap> Y:=ContractedComplex(Y);;
gap> R:=PureComplexRandomCell(Y);;
gap> D:=PureComplexDifference(Y,R);;
gap> D:=ContractedComplex(D);;
gap> Display(D);
gap> Y2:=ContractedComplex(Y2);;

gap> Size(FiltrationTerm(F,10));
918881
gap> Size(Y2); 
61618

gap> Y1:=PureComplexDifference(Y2,PureComplexRandomCell(Y2));;
gap> Y1:=ContractedComplex(Y1);;
gap> Size(Y1);
474
gap> Display(Y1);



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31 changes: 19 additions & 12 deletions tutorial/chap5_mj.html
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5_mj.html#X86FD0A867EC9E64F">5.5-1 <span class="Heading">Non-trivial cup product</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5_mj.html#X802C7755851A2530">5.5-2 <span class="Heading">Explicit first homology generators</span></a>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap5_mj.html#X783EF0F17B629C46">5.5-2 <span class="Heading">Explicit homology generators</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap5_mj.html#X80D0D8EB7BCD05E9">5.6 <span class="Heading">Knotted proteins</span></a>
Expand Down Expand Up @@ -323,23 +323,30 @@ <h5>5.5-1 <span class="Heading">Non-trivial cup product</span></h5>

</pre></div>

<p><a id="X802C7755851A2530" name="X802C7755851A2530"></a></p>
<p><a id="X783EF0F17B629C46" name="X783EF0F17B629C46"></a></p>

<h5>5.5-2 <span class="Heading">Explicit first homology generators</span></h5>
<h5>5.5-2 <span class="Heading">Explicit homology generators</span></h5>

<p>It could be desirable to obtain explicit representatives of the two homology generators in degree <span class="SimpleMath">\(1\)</span> that "persist". The following commands visualize the two explicit homology generators by visualizing a subspace <span class="SimpleMath">\(D\subset X_{10}\)</span> where <span class="SimpleMath">\(D\)</span> is homotopy equivalent to a wedge of two circles and where the inclusion induces an isomorphism on first homology groups.</p>
<p>It could be desirable to obtain explicit representatives of the persistent homology generators that "persist" through a significant sequence of filtration terms. There are two such generators in degree <span class="SimpleMath">\(1\)</span> and one such generator in degree <span class="SimpleMath">\(2\)</span>. The explicit representatives in degree <span class="SimpleMath">\(n\)</span> could consist of an inclusion of pure cubical complexes <span class="SimpleMath">\(Y_n \subset X_{10}\)</span> for which the incuced homology homomorphism <span class="SimpleMath">\(H_n(Y_n,\mathbb Z) \rightarrow H_n(X_{10},\mathbb Z)\)</span> is an isomorphism, and for which <span class="SimpleMath">\(Y_n\)</span> is minimal in the sense that its homotopy type changes if any one or more of its top dimensional cells are removed. Ideally the space <span class="SimpleMath">\(Y_n\)</span> should be as "close to the original dataset" <span class="SimpleMath">\(X_0\)</span>. The following commands first construct an explicit degree <span class="SimpleMath">\(2\)</span> homology generator representative <span class="SimpleMath">\(Y_2\subset X_{10}\)</span> where <span class="SimpleMath">\(Y_2\)</span> is homotopy equivalent to <span class="SimpleMath">\(X_{10}\)</span>. They then construct an explicit degree <span class="SimpleMath">\(1\)</span> homology generators representative <span class="SimpleMath">\(Y_1\subset X_{10}\)</span> where <span class="SimpleMath">\(Y_1\)</span> is homotopy equivalent to a wedge of two circles. The final command displays the homology generators representative <span class="SimpleMath">\(Y_1\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Y:=FiltrationTerm(F,10);; </span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">for n in Reversed([1..9]) do</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">Y:=ContractedComplex(Y,FiltrationTerm(F,n));</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Y2:=FiltrationTerm(F,10);; </span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">for t in Reversed([1..9]) do</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">Y2:=ContractedComplex(Y2,FiltrationTerm(F,t));</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">od;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Y:=ContractedComplex(Y);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">R:=PureComplexRandomCell(Y);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">D:=PureComplexDifference(Y,R);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">D:=ContractedComplex(D);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Display(D);</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Y2:=ContractedComplex(Y2);;</span>

<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(FiltrationTerm(F,10));</span>
918881
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(Y2); </span>
61618

<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Y1:=PureComplexDifference(Y2,PureComplexRandomCell(Y2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Y1:=ContractedComplex(Y1);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(Y1);</span>
474
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Display(Y1);</span>

</pre></div>

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