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products/sources/formale Sprachen/GAP/pkg/hap/www/SideLinks/About/ (Algebra von RWTH Aachen Version 4.15.1^©) Datei vom 19.6.2025 mit Größe 9 kB

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products/sources/formale Sprachen/GAP/pkg/hap/www/SideLinks/About/aboutPeripheral.html

<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
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content="text/html; charset=ISO-8859-1">
 <title>AboutHap</title>
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style="color: rgb(0, 0, 153); background-color: rgb(204, 255, 255);"
link="#000066" vlink="#000066" alink="#000066">
 
<table
style="text-align: left; margin-left: auto; margin-right: auto; color: rgb(0, 0, 102);"
border="0" cellpadding="20" cellspacing="10">
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 <tr align="center">
 <th style="vertical-align: top;">
 <table style="width: 100%; text-align: left;" cellpadding="2"
cellspacing="2">
 <tbody>
 <tr>
 <td style="vertical-align: top;"><a href="aboutKnots.html">Previous</a> 
 </td>
 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: Computing a peripheral system
for a 3-manifold 
 </big></td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutCoverinSpaces.html">next</a> 
 </td>
 </tr>
 </tbody>
 </table>
 <big></big> 
 </th>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">Let
K:S1-->R3
be a tame knot. Let TK be a solid tubular neighbourhood of the knot,
the neighbourhood being small enough to have the homotopy type of a
circle. Let MK denote the closure of the complement R3\TK . Then MK
is a 3-manifold. By a theorem of Waldhausen [F. Waldhausen, "On
irreducible 3-manifolds which are sufficiently large", Annals of
Mathematics. Second Series 87 (1968), 56–88] the
homeomorphism type of MK is completely determined by
the canonical inclusion of fundamental groups π1(δMK)
-->
π1(MK) where δMK denotes the
boundary of MK . This homomorphism is an example of a peripheral system. By a theorem of
Gordon
and Luecke [C. Gordon and J. Luecke, "Knots are determined by their
Complements", J. Amer. Math. Soc., 2 (1989), 371–415]
the homeomorphism type of MK completely determines the
ambient isotopy type of the knot K. 
 
As a means of illustrating some HAP functions for computing with
topological manifolds we shall compute the homomorphism π1(δMK)
-->
π1(MK) for
the T.thermophilus 1V2X protein knot illustrated on the <a
href="aboutKnots.html#proteins">previous page</a>. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands first read the protein knot as a pure permutahedrall
complex from a <a
href="1V2X.pdb">pdb
file</a> . 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
K:=ReadPDBfileAsPurePermutahedralComplex("1V2X.pdb"); 
Reading chain containing 191 atoms. 
Pure permutahedral complex of dimension 3. 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 255); vertical-align: top;">The
advantage of working with pure permutahedral complexes is that they are
always topological manifolds. (This is not the case for pure cubical
complexes.) 
 
The following commands compute a pure permutahedral complex M
homeomorphic to the manifold MK. The complex M is a union of
4433 3-dimensional permutahedra. 
 </td>
 </tr>
 <tr>
 <td
style="background-color: rgb(255, 255, 204); vertical-align: top;">gap>
K:=ZigZagContractedPureComplex(K); 
Pure permutahedral complex of dimension 3. 
 
gap> K:=PurePermutahedralComplex(FrameArray(K!.binaryArray)); 
Pure permutahedral complex of dimension 3. 
 
gap> M:=ComplementOfPureComplex(K); 
Pure permutahedral complex of dimension 3. 
 
gap> M:=ZigZagContractedPureComplex(M); 
Pure permutahedral complex of dimension 3. 
 
gap> Size(M); 
4433 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
next command converts M to a homeomorphic regular CW-complex Y with the
same cellular structure as M. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
Y:=PermutahedralComplexToRegularCWComplex(M); 
Regular CW-complex of dimension 3 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">(The
next
commands
are
not
needed
for the computation of the homomorphism π1(δMK)
-->
π1(MK) . They produce a regular CW-complex W
which is
homeomorphic to Y but has fewer cells than Y. The manifold Y has a 4433
3-dimensional cells. The manifold W has just 32 cells of dimension 3. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
Y!.nrCells(3); 
4433 
 
gap> W:=SimplifiedRegularCWComplex(Y); 
Regular CW-complex of dimension 3 
 
gap> W!.nrCells(3); 
32 
 
gap>#

) 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">We
now compute the boundary B of Y. This boundary will have two path
components: one will be a surface homeomorphic to a torus, the other
will be homeomorphic to a 2-sphere. 
 
By computing the fundamental groups of B based at two different 0-cells
we observe that the 0-cell numbered 35296 lies in the torus. 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
B:=BoundaryOfPureComplex(Y); 
Regular CW-complex of dimension 2 
 
gap> CriticalCellsOfRegularCWComplex(B); 
[ [ 2, 1 ], [ 2, 1089 ], [ 1, 3575 ], [ 1, 58055 ], [ 0, 29938 ], [ 0,
35296 ] ] 
 
gap> F:=FundamentalGroup(f,29938); 
[ <identity ...> ] -> [ <identity ...> ] 
 
gap> F:=FundamentalGroup(f,35296); 
[ f1, f2 ] -> [ f1^-1*f2^3*f1, f2^-1 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
next commands compute a finite presentation for π1Y . 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
pi:=FundamentalGroup(Y); 
<fp group of size infinity on the generators [ f1, f2 ]> 
 
gap> RelatorsOfFpGroup(pi); 
[ f1*f2*f1^-1*f2*f1 ] 
 </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">In
summary: 
 
 <table
style="text-align: left; width: 40%; margin-left: auto; margin-right: auto;"
border="0" cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td style="vertical-align: top;">π1Y = < x,
y | xyx-1yx > 
 
π1B = < u, v | uvu-1v-1
> 
 
π1B ----> π1Y, u --> x-1y3x
,
v
->
y-1 
 </td>
 </tr>
 </tbody>
 </table>
 
 </td>
 </tr>
 <tr>
 <td style="vertical-align: top;">
 <table
style="margin-left: auto; margin-right: auto; width: 100%; text-align: left;"
border="0" cellpadding="2" cellspacing="2">
 <tbody>
 <tr>
 <td style="vertical-align: top;"><a
style="color: rgb(0, 0, 102);" href="aboutKnots.html">Previous
Page</a> 
 </td>
 <td style="text-align: center; vertical-align: top;"><a
href="aboutContents.html">Contents</a> 
 </td>
 <td style="text-align: right; vertical-align: top;"><a
href="aboutCoverinSpaces.html">Next
page 
 </a> </td>
 </tr>
 </tbody>
 </table>
 <a href="aboutTopology.html"> 
 </a> </td>
 </tr>
 </tbody>
</table>
 
 
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</html>

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