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 A-polynomial 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 A-polynomial. 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.

48 pages, 9 figures. Comments are welcome

References in corpus (9)

• Floer homology and knot complements
• Open string amplitudes and large order behavior in topological string theory
• On the characteristic and deformation varieties of a knot
• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Super-A-polynomial for knots and BPS states
• Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots
• 3d analogs of Argyres-Douglas theories and knot homologies
• Khovanov Homology and Conway Mutation
• Knot Floer homology, genus bounds, and mutation

Cited by in corpus (11)

• Walls, Lines, and Spectral Dualities in 3d Gauge Theories
• SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry
• A note on colored HOMFLY polynomials for hyperbolic knots from WZW models
• 4d Quantum Geometry from 3d Supersymmetric Gauge Theory and Holomorphic Block
• The colored HOMFLYPT function is $q$-holonomic
• The colored HOMFLY polynomial is q-holonomic
• Condensates and instanton - torus knot duality. Hidden Physics at UV scale
• Instanton-torus knot duality in 5d SQED and $SU(2)$ SQCD
• Homfly polynomials for periodic knots via state model
• A generating series for Murakami-Ohtsuki-Yamada graph evaluations
• Vassiliev Invariants for Flows Via Chern-Simons Perturbation Theory