Lectures on Knot Homology and Quantum Curves
Abstract
Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this interpretation allows one to pose questions that would not have been asked otherwise, such as, "Is there a direct relation between Khovanov homology and the Apolynomial of a knot?" We will explain that the answer to this question is "yes," and introduce a certain deformation of the planar algebraic curve defined by the zero locus of the Apolynomial. This novel deformation leads to a categorified version of the Generalized Volume Conjecture that completely describes the "color behavior" of the colored sl(2) knot homology, and eventually to a similar version for the colored HOMFLY homology. Furthermore, this deformation is strong enough to distinguish mutants, and its most interesting properties include relations to knot contact homology and knot Floer homology.
 Publication:

arXiv eprints
 Pub Date:
 November 2012
 arXiv:
 arXiv:1211.6075
 Bibcode:
 2012arXiv1211.6075G
 Keywords:

 High Energy Physics  Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  Geometric Topology;
 Mathematics  Quantum Algebra
 EPrint:
 48 pages, 9 figures. Comments are welcome