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2003•Journal of Knot Theory and Its RamificationsRequires access

THE NONCOMMUTATIVE A-IDEAL OF A (2, 2p + 1)-TORUS KNOT DETERMINES ITS JONES POLYNOMIAL

Răzvan Gelca, Jeremy Sain

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Abstract

The noncommutative A-ideal of a knot is a generalization of the A-polynomial, defined using Kauffman bracket skein modules. In this paper we show that any knot that has the same noncommutative A-ideal as the (2,2p + 1)-torus knot has the same colored Jones polynomials. This is a consequence of the orthogonality relation, which yields a recursive relation for computing all colored Jones polynomials of the knot.

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What this paper is about

The noncommutative A-ideal of a knot is a generalization of the A-polynomial, defined using Kauffman bracket skein modules. In this paper we show that any knot that has the same noncommutative A-ideal as the (2,2p + 1)-torus knot has the same colored Jones polynomials. This is a consequence of the orthogonality relation, which yields a recursive relation for computing all colored Jones polynomials of the knot.

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OpenAlex reports 43 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

The noncommutative A-ideal of a knot is a generalization of the A-polynomial, defined using Kauffman bracket skein modules. In this paper we show that any knot that has the same noncommutative A-ideal as the (2,2p + 1)-torus knot has the same colored Jones polynomials. This is a consequence of the orthogonality relation, which yields a recursive relation for computing all colored Jones polynomials of the knot.

Key concepts: Knot polynomial, Mathematics, Quantum invariant, Bracket polynomial, Noncommutative geometry, Skein relation, Knot (papermaking), Jones polynomial

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