The C–polynomial of a knot

Garoufalidis, Stavros¹ ; Sun, Xinyu²

¹School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA
²Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX 77843-3368, USA

Algebraic and Geometric Topology, Tome 6 (2006) no. 4, pp. 1623-1653

Résumé

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 Uq(sl2)) specializes at q = 1 to the better known A–polynomial of a knot, which has to do with genuine SL2(ℂ) 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.

Détail

DOI : 10.2140/agt.2006.6.1623

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

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