Permutation Jones Polynomials

We introduce a generalization of the Jones polynomial for classical and virtual knots and links using colorings by a permutation $\sigma:X\to X$ of a finite set $X$. For $X=\{1\}$ and for classical knots, the invariant is equivalent to the usual Jones polynomial; for $X$ with cardinality greater than 1 the invariant expresses distinct information from the Jones polynomial or virtual knots and for classical and virtual links. We establish some properties of the new invariants and compute the polynomials for classical and virtual knots and links of small crossing number for a few small permutations.