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<
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big><
span
style=
"font-weight: bold;">About HAP: The fundamental group<
br>
</
span></
big></
td>
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href=
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small style=
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<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">As
an
illustration
of HAP
's functions for cellular spaces we use them to
obtain a presentation for the fundamental group of the complement of
the
Borromean rings
link in Euclidean 3-space. (The method uses no
specialized techniques from knot theory and can be applied to fairly
arbitrary subspaces of Euclidean space.)<
br>
<
div style=
"text-align: center;"><
br>
<
img style=
"width: 300px; height: 249px;" alt=
""
src=
"borromean-rings.jpg"><
br>
<
br>
</
div>
</
td>
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<
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<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">The
function <
big><
span style=
"font-family: monospace;">ReadLinkImageAsPureCubicalComplex(
"link.eps")</
span></
big>
inputs an image file containing a
link diagram and tries to
output the
corresponding
link as a 3-dimensional pure cubical complex. Ideally the
link diagram should have dimensions (line thickness, under-crossing
spaces etc) similar to that of the Borromean rings diagram above. The
function does not always succeed: it applies several checks, and if one
of these checks fails then the function returns
"fail".<
br>
<
br>
The following commands succeed in using the above image to realize the
the Borromean rings as
a pure cubical complex with 33654 3-dimensional cubes.<
br>
</
td>
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<
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<
td
style=
"background-color: rgb(255, 255, 204); vertical-align: top;">gap>
L:=ReadLinkImageAsPureCubicalComplex(
"borromean-rings.jpg");<
br>
Pure cubical complex of dimension 3.<
br>
<
br>
gap> Size(L);<
br>
33654<
br>
</
td>
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<
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<
td
style=
"background-color: rgb(255, 255, 255); vertical-align: top;">The
following commands confirm that L has the correct homology.<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"background-color: rgb(255, 255, 204); vertical-align: top;">gap>
Homology(L,0);<
br>
[ 0, 0, 0 ]<
br>
<
br>
gap> Homology(L,1);<
br>
[ 0, 0, 0 ]<
br>
<
br>
gap> Homology(L,2);<
br>
[ ]<
br>
</
td>
</
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<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands produce a 3-dimensional pure cubical complex C that
is homotopy equivalent to the complement of the Borromean rings in
Euclidean 3-space. The complex C involves 116123 3-dimensional cubes.<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
C:=ComplementOfPureCubicalComplex(L);<
br>
Pure cubical complex of dimension 3.<
br>
<
br>
gap> C:=ZigZagContractedPureCubicalComplex(C);<
br>
Pure cubical complex of dimension 3.<
br>
<
br>
gap> Size(C);<
br>
116123<
br>
</
td>
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<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">The
next commands convert C to a regular CW-space Y involving a total of
1447707 cells of dimension 0,1,2 and 3. The CW-structure on Y is
precisely that of the pure cubical complex C.<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
Y:=CubicalComplexToRegularCWSpace(C);<
br>
Regular CW-space of dimension 3<
br>
<
br>
gap> Size(Y);<
br>
1447707<
br>
</
td>
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<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">Next
we
construct
a discrete vector field on Y which has just 9 critical
cells, three in dimension 2, four in dimension 1, and two zero cells. <
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
CriticalCellsOfRegularCWSpace(Y);<
br>
[ [ 2, 334 ], [ 2, 115000 ], [ 2, 139630 ], [ 1, 386713 ], [ 1, 404957
],<
br>
[ 1, 405056 ], [ 1, 600331 ], [ 0, 164802 ], [ 0, 241782 ] ]<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">Discrete
Morse
Theory
tells us that there exists a CW-space W that is homotopy
equivalent to Y (and C) and that has just three 2-cells, four 1-cells
and two 0-cells. The space W can be used to find a presentation of
the fundamental group of Y and to determine the homology of Y. <
br>
<
br>
The following commands use W to confirm that the
link complement Y has
the correct integral homology.<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
Homology(Y,0);<
br>
[ 0 ]<
br>
gap> Homology(Y,1);<
br>
[ 0, 0, 0 ]<
br>
gap> Homology(Y,2);<
br>
[ 0, 0, 0 ]<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands use W to determine a presentation for the
fundamental group of the
link complement Y.<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
F:=FundamentalGroup(Y);<
br>
<fp group on the generators [ f1, f2, f3 ]><
br>
<
br>
gap> RelatorsOfFpGroup(simplifiedF);<
br>
[ f3^-1*f2^-1*f3*f1^-1*f2*f3*f2^-1*f3^-1*f1*f2,<
br>
f2^-1*f3^-1*f1*f3*f1^-1*f2*f1*f3^-1*f1^-1*f3 ] <
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">Note
that
the
abelianization F<
sub>ab</
sub> is free abelian of rank 3.
This is consistent with the isomorphism H<
sub>1</
sub>(Y,Z) = H<
sub>1</
sub>(F,Z).<
br>
</
td>
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tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">The
fundamental group F can be used to compute some
link invariants. For
instance, we can compute the degree 3 integral homology of its class 2
nilpotent quotient G:=F/[[F,F],F] .<
br>
</
td>
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tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
G:=NilpotentQuotient(F,3);;<
br>
<
br>
gap> GroupHomology(G,3);<
br>
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0,<
br>
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]<
br>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands compute the same homological invariant for the
following
link<
br>
<
br>
<
div style=
"text-align: center;"><
img
style=
"width: 310px; height: 215px;" alt=
"" src=
"link.jpg"> <
br>
<
div style=
"text-align: left;"><
br>
and, as the value of the invariant is different in this case, the
computations prove that
the Borromean rings are not ambient isotopic to this second
link.<
br>
</
div>
</
div>
</
td>
</
tr>
<
tr>
<
td
style=
"vertical-align: top; background-color: rgb(255, 255, 204);">gap>
L:=ReadLinkImageAsPureCubicalComplex(
"link.jpg");;<
br>
gap> C:=ComplementOfPureCubicalComplex(L);;<
br>
gap> C:=ZigZagContractedPureCubicalComplex(C);;<
br>
gap> Y:=CubicalComplexToRegularCWSpace(C);;<
br>
gap> F:=FundamentalGroup(Y);<
br>
gap> G:=NilpotentQuotient(F,3);;<
br>
gap> GroupHomology(G,3);<
br>
[ 0, 0, 0, 0, 0, 0 ]<
br>
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