About HAP: Computing a peripheral system for a 3-manifold

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.
The following commands first read the protein knot as a pure permutahedrall complex from a pdb file
gap> K:=ReadPDBfileAsPurePermutahedralComplex("1V2X.pdb");
Reading chain containing 191 atoms.
Pure permutahedral complex of dimension 3.
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.
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);
The next command converts M to a homeomorphic regular CW-complex Y with the same cellular structure as M.
gap> Y:=PermutahedralComplexToRegularCWComplex(M);
Regular CW-complex of dimension 3
(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.
gap> Y!.nrCells(3);

gap> W:=SimplifiedRegularCWComplex(Y);
Regular CW-complex of dimension 3

gap> W!.nrCells(3);

gap>#                                                                   )
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.
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 ]
The next commands compute a finite presentation for π1Y .
gap> pi:=FundamentalGroup(Y);
<fp group of size infinity on the generators [ f1, f2 ]>

gap> RelatorsOfFpGroup(pi);
[ f1*f2*f1^-1*f2*f1 ]
In summary:

π1Y = < x, y  |  xyx-1yx >

π1B = < u, v  |  uvu-1v-1 >

π1B ---->  π1Y,  u --> x-1y3x ,  v -> y-1
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