The $C$–polynomial of a knot

Abstract

In an earlier paper the first author defined a non-commutative $A$–polynomial for knots in 3–space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear $q$–difference equation. Said differently, the colored Jones function of a knot is annihilated by a non-zero ideal of the Weyl algebra which is generalted (after localization) by the non-commutative $A$–polynomial of a knot.

In that paper, it was conjectured that this polynomial (which has to do with representations of the quantum group $U_q\left({\mathfrak{s}\mathfrak{l}}_2\right)$) specializes at $q=1$ to the better known $A$–polynomial of a knot, which has to do with genuine ${SL}_2\left(ℂ\right)$ representations of the knot complement.

Computing the non-commutative $A$–polynomial of a knot is a difficult task which so far has been achieved for the two simplest knots. In the present paper, we introduce the $C$–polynomial of a knot, along with its non-commutative version, and give an explicit computation for all twist knots. In a forthcoming paper, we will use this information to compute the non-commutative $A$–polynomial of twist knots. Finally, we formulate a number of conjectures relating the $A$, the $C$–polynomial and the Alexander polynomial, all confirmed for the class of twist knots.