Torsion Numbers Knot

The first homology of the k-fold branched cover of a knot, like all finite abelian groups, can be described uniquely as a finite direct sum Z/n_1(Z) + Z/n_2(Z) + ...., where each n_i divides the next. The sequence of n_i are the "torsion numbers" of the knot. A basic theorem states that they are all relatively prime to k. For k =2, their product is the determinant of the knot.

As an example, the knots 8_18 and 9_23 both have determinant 45. The homology of the two-fold cover of 8_18 is Z_3 + Z+15, while that of 9_23 is Z_45. These are denoted {3,15} and {45}.

The trivial group is denoted {1}.

The table presents the torsion numbers for k from 2 to 9. For instance for 4_1 we have:

{{2,{5}},{3,{4,4}},{4,{3,15}},{5,{11,11}},{6,{8,40}},{7,{29,29}},{8,{21,105}},{9,{76,76}}}

So the 8-fold branched cover of 4_1 has homology Z_21 + Z_105 and the 9-fold cover has homology Z_76 + Z_76.
A presentation matrix for the homology of the n-fold branched cover of a knot K with Seifert matrix V is as follows:

Let G = (V^t - V)^{-1} V^t.

The presentation matrix is G^n - (G - I)^n.