The C–polynomial of a knot

Garoufalidis, Stavros; Sun, Xinyu

doi:10.2140/agt.2006.6.1623

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The $C$–polynomial of a knot

Stavros Garoufalidis and Xinyu Sun

Algebraic & Geometric Topology 6 (2006) 1623–1653

DOI: 10.2140/agt.2006.6.1623

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 $U_{q} ({s l}_{2})$ ) specializes at $q = 1$ to the better known $A$ –polynomial of a knot, which has to do with genuine ${SL}_{2} (ℂ)$ 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.

Keywords

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

Mathematical Subject Classification 2000

Primary: 57N10

Secondary: 57M25

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Publication

Received: 9 June 2005

Revised: 4 August 2006

Accepted: 29 August 2006

Published: 11 October 2006

Authors

Stavros Garoufalidis
School of Mathematics Georgia Institute of Technology Atlanta, GA 30332-0160 USA
http://www.math.gatech.edu/~stavros
Xinyu Sun
Department of Mathematics Mailstop 3368 Texas A&M University College Station, TX 77843-3368 USA
http://www.math.tamu.edu/~xsun/

Volume 6, issue 4 (2006)