On the relation between the A-polynomial and the Jones polynomial

By R. Gelca. 2000

56 citations recorded by the source corpus.

This paper shows that the noncommutative generalization of the A-polynomial of a knot, defined using Kauffman bracket skein modules, together with finitely many colored Jones polynomials, determines the remaining colored Jones polynomials of the knot. It also shows that under certain conditions, satisfied for example by the unknot and the trefoil knot, the noncommutative generalization of the A-polynomial determines all colored Jones polynomials of the knot.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7b590346b07f74169a4c433786f9f8b675336c0a

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In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

On the characteristic and deformation varieties of a knot (2003, background): “This can be converted into a recursion relation using forward shifts by: In[6]:= ForwardShifts[% ] 3 + 2 n 2 2 + n q (q - q ) SUM[n] Out[6]= ----------------------------- + 3 4 + 2 n q - q -2 - n 2 + n 2 + n > (q (q - q ) (q + q ) 4 5 + n 6 + 2 n 7 + 2 n 7 + 3 n 8 + 4 n Geometry & Topology Monograph”
The colored Jones polynomial and the A-polynomial of Knots☆ (2004, result): “Partial results were obtained earlier by Frohman, Gelca, and Lofaro through their theory of non-commutative A-ideal [FGL, Ge], which also plays an important role in the present paper.”
Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial (2003, background): “3) It would be interesting to understand a relation to the work [86], where a similar connection between the A-polynomial and the Jones polynomial was proposed from the the non-commutative point of view.”
Quantum Riemann Surfaces in Chern-Simons Theory (2011, unknown): “in [7, 43, 44] and later [45].”