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| 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 previous page. |
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| The
following commands first read the protein knot as a pure permutahedrall
complex from a pdb
file . |
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| gap>
K:=ReadPDBfileAsPurePermutahedralComplex("1V2X.pdb"); Reading chain containing 191 atoms. Pure permutahedral complex of dimension 3. |
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| 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. |
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| 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 |
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| The
next command converts M to a homeomorphic regular CW-complex Y with the
same cellular structure as M. |
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| gap>
Y:=PermutahedralComplexToRegularCWComplex(M); Regular CW-complex of dimension 3 |
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| (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. |
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| gap>
Y!.nrCells(3); 4433 gap> W:=SimplifiedRegularCWComplex(Y); Regular CW-complex of dimension 3 gap> W!.nrCells(3); 32 gap># ) |
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| 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. |
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| 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 ] |
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| The
next commands compute a finite presentation for π1Y . |
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| gap>
pi:=FundamentalGroup(Y); <fp group of size infinity on the generators [ f1, f2 ]> gap> RelatorsOfFpGroup(pi); [ f1*f2*f1^-1*f2*f1 ] |
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In
summary:
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