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.
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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.
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