Torsion Numbers

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/n1(Z) + Z/n2(Z) + ...., where each ni divides the next. The sequence of ni 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 818 and 923 both have determinant 45. The homology of the two-fold cover of 818 is Z3 + Z15, while that of 923 is Z45. 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 41 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 41 has homology Z21 + Z105 and the 9-fold cover has homology Z76 + Z76.

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 = (Vt - V){-1} Vt.
The presentation matrix is Gn - (G - I)n.