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 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 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 + 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 Gⁿ - (G - I)^n.