Volume 6, issue 4 (2006)

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The $C$–polynomial of a knot

Stavros Garoufalidis and Xinyu Sun

Algebraic & Geometric Topology 6 (2006) 1623–1653
 arXiv: math.GT/0504305
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.

Keywords
WZ algorithm, creative telescoping, colored Jones function, Gosper's algorithm, cyclotomic function, holonomic functions, characteristic varieties, $A$-polynomial, $C$-polynomial
Primary: 57N10
Secondary: 57M25