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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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 Uq(sl2)) specializes at q = 1 to the better known A–polynomial of a knot, which has to do with genuine SL2() 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
References
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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/