The C-polynomial of a knot

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dc.creatorGaroufalidis, Stavros
dc.creatorSun, Xinyu
dc.date2005-04-14
dc.date2009-04-30
dc.date.accessioned2026-07-07T13:10:02Z
dc.date.available2026-07-07T13:10:02Z
dc.descriptionIn 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(SL_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.
dc.descriptionThis is the version published by Algebraic & Geometric Topology on 11 October 2006
dc.identifierhttps://arxiv.org/abs/math/0504305
dc.identifierhttp://arxiv.org/abs/math/0504305
dc.identifierAlgebr. Geom. Topol. 6 (2006) 1623-1653
dc.identifierdoi:10.2140/agt.2006.6.1623
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/228935
dc.subjectGeometric Topology
dc.subjectCombinatorics
dc.subject57N10, 57M25
dc.titleThe C-polynomial of a knot
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