Narrow Results

In this paper, we first investigate the minimal sufficient sets of colors for $p=11$ and $13$. For odd prime $p$ and any $p$-colorable link $L$ with $detL\ne 0$, we give alternative proofs of ${\mathrm{\text{mincol}}}_{p}L\ge 5$ for $p\ge 11$ and ${\mathrm{\text{mincol}}}_{p}L\ge 6$ for $p\ge 17$. We also elaborate on equivalence classes of sets of distinct colors (on a given modulus) and prove that there are two such classes of five colors modulo $11$, and only one such class of five colors modulo 13.

If $A$ is an abelian group and $\varphi$ is an integer, let $A(\varphi )$ be the subgroup of $A$ consisting of elements $a\in A$ such that $\varphi \cdot a=0$. We prove that if $D$ is a diagram of a classical link $L$ and $0={\varphi }_0,{\varphi }_1,\dots ,{\varphi }_{n-1}$ are the invariant factors of an adjusted Goeritz matrix of $D$, then the group ${{𝒟}}_A(D)$ of Dehn colorings of $D$ with values in $A$ is isomorphic to the direct product of $A$ and $A=A({\varphi }_0),A({\varphi }_1),\dots ,A({\varphi }_{n-1})$. It follows that the Dehn coloring groups of $L$ are isomorphic to those of a connected sum of torus links $T_{(2,{\varphi }_1)}\#\cdots \#T_{(2,{\varphi }_{n-1})}$.

In a recent paper, Jones introduced a correspondence between elements of the Thompson group $F$ and certain graphs/links. It follows from his work that several polynomial invariants of links, such as the Kauffman bracket, can be reinterpreted as coefficients of certain unitary representations of $F$. We give a somewhat different and elementary proof of this fact for the Kauffman bracket evaluated at certain roots of unity by means of a statistical mechanics model interpretation. Moreover, by similar methods we show that, for some particular specializations of the variables, other familiar link invariants and graph polynomials, namely the number of $N$-colorings and the Tutte polynomial, can be viewed as positive definite functions on $F$.