Equations on knot polynomials and 3d/5d duality
Abstract
We briefly review the current situation with various relations between knot/braid polynomials (ChernSimons correlation functions), ordinary and extended, considered as functions of the representation and of the knot topology. These include linear skein relations, quadratic Plucker relations, as well as "differential" and (quantum) ̂Apolynomial structures. We pay a special attention to identity between the ̂Apolynomial equations for knots and Baxter equations for quantum relativistic integrable systems, related through SeibergWitten theory to 5d superYangMills models and through the AGT relation to the qVirasoro algebra. This identity is an important ingredient of emerging a 3d  5d generalization of the AGT relation. The shape of the Baxter equation (including the values of coefficients) depend on the choice of the knot/braid. Thus, like the case of KP integrability, where (some, so far torus) knots parameterize particular points of the Universal Grassmannian, in this relation they parameterize particular points in the moduli space of manybody integrable systems of relativistic type.
 Publication:

The Sixth International School on Field Theory and Gravitation2012
 Pub Date:
 October 2012
 DOI:
 10.1063/1.4756970
 arXiv:
 arXiv:1208.2282
 Bibcode:
 2012AIPC.1483..189M
 Keywords:

 algebra;
 ChernSimons theory;
 polynomials;
 YangMills theory;
 02.10.v;
 02.10.De;
 11.15.Yc;
 Logic set theory and algebra;
 Algebraic structures and number theory;
 High Energy Physics  Theory;
 Mathematics  Geometric Topology;
 Mathematics  Quantum Algebra
 EPrint:
 19 pages, contribution to Proceedings of 6th International School on Field Theory and Gravitation (Petropolis, Brazil, 2012)