For a link $L$ and a prime $p$, $L$ has a $p$-coloring iff $p$ divides the $\operatorname{gcd}$ of the invariant factors of the Goeritz matrix of $L$.
Do you know the elementary proof of this facts?
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$\begingroup$ Please use TeX. Besides, more details and definitions would help. $\endgroup$– Alex DegtyarevCommented Nov 12, 2017 at 15:32
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1 Answer
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A very nice treatment of this is given in Chapter 9 of Lickorish's book, "An introduction to knot theory". He explains that the Goeritz matrix is a presentation matrix for the first homology of the double branched cover of the knot. It is relatively straightforward to prove that there is a $p$-coloring iff there is a dihedral representation of the knot group iff $p$ divides the order of that first homology. The latter equivalence uses the fact that the covering transformation acts by $-1$ on the first homology of the double branched cover.
answered Nov 12, 2017 at 20:46
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$\begingroup$ Moreover, isn't it true that the dimension of the $\mathbb{F}_p$-vector space of p-colorings of a knot is equal to the dimension of $\mathbb{F}_p \otimes M$, where $M$ denotes the first homology of the double branched covering? $\endgroup$ Commented Nov 14, 2017 at 10:14
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$\begingroup$ Dear Prof. Danny Ruberman, thank you very much for your reply. $\endgroup$ Commented Nov 15, 2017 at 3:24