These algorithms, as well as previously implemented methods for lifting chain maps, allow for computation of (co)homology with local coeffeicients. Please see this preprint (submitted to Journal of Symbolic Computation) for a more detailed treatment of this (and the below) processes, as well as a description of a certain knot invariant which can be obtained from a finite-index covering of a knot complement. The below functions are included in HAP as of version 1.22.

Complements/knotcomp.g

KnotComplement will input an arc presentation representing some knot and output a regular CW-decomposition of its complement. The below code computes the complement of the trefoil, displays the number of cells in the complex and obtains its 0^th, 1^st & 2^nd homology groups.

gap> arc:=ArcPresentation(PureCubicalKnot(3,1));
[ [ 2, 5 ], [ 1, 3 ], [ 2, 4 ], [ 3, 5 ], [ 1, 4 ] ]
gap> comp:=KnotComplement(arc);
Regular CW-complex of dimension 3

gap> Size(comp);
395
gap> for i in [0..2] do Print(Homology(comp,i),"\n"); od;
[ 0 ]
[ 0 ]
[ 0 ]

An optional argument of "rand" will enable random generation of 2-cells. This change results in the same complex, but the alternate ordering may offer a different (but equivalent) presentation of the fundamental group.

gap> pi_1:=FundamentalGroup(comp,1);
#I  there are 2 generators and 1 relator of total length 6
<fp group of size infinity on the generators [ f1, f2 ]>
gap> RelatorsOfFpGroup(pi_1);
[ f1*f2^-1*f1^-1*f2*f1^-1*f2^-1 ]
gap> comp:=KnotComplement(arc,"rand");
Random 2-cell selection is enabled.
Regular CW-complex of dimension 3

gap> pi_1:=FundamentalGroup(comp,1);; RelatorsOfFpGroup(pi_1);
#I  there are 2 generators and 1 relator of total length 6
[ f2^-1*f1*f2^-1*f1^-1*f2*f1^-1 ]

Complements/knotcompbound.g

KnotComplementWithBoundary takes an arc presentation as above and returns an inclusion map of regular CW-complexes from the boundary of a knot to its complement. The below code calculates this inclusion from the 3rd prime knot on 11 crossings. The induced homomorphism from their fundamental groups is then computed.

gap> arc:=ArcPresentation(PureCubicalKnot(11,3));
[ [ 3, 9 ], [ 2, 5 ], [ 1, 3 ], [ 7, 10 ], [ 4, 8 ], [ 6, 9 ], [ 7, 11 ], 
  [ 5, 10 ], [ 2, 6 ], [ 4, 11 ], [ 1, 8 ] ]
gap> i:=KnotComplementWithBoundary(arc);
Map of regular CW-complexes

gap> FundamentalGroup(i,1);
[ f1, f2 ] -> 
[ 
  f2*f1*f2^-1*f1*f3^-1*f2^-1*f3*f2*f3*f1^-1*f3*f2*f3^-1*f2^-1*f3^-1*f2*f3*f2^-1\
*f1*f3^-1*f2^-1*f3*f2*f3*f1^-1*f2*(f1*f2^-1*f3^-1)^3*f1*f3^-1*f2^-1*f3^-1*f2*f\
3*f1^-1*f3^-1*f2^-1*(f3*f2)^2*f3*f1^-1, f1*f3^-1*f2^-1*f3^-1*f2*f3*f1^-1 ]