Lectures on knot homology and quantum curves. (English) Zbl 1320.81084
Stolz, Stephan (ed.), Topology and field theories. Center for Mathematics at Notre Dame. Summer school and conference, University of Notre Dame, Notre Dame, IN, USA, May 29 – June 8, 2012. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-1015-5/pbk; 978-1-4704-1552-5/ebook). Contemporary Mathematics 613, 41-78 (2014).
Summary: 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 \(\mathfrak{sl}(2)\) 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.
For the entire collection see [Zbl 1287.00023].
For the entire collection see [Zbl 1287.00023].
MSC:
81T45 | Topological field theories in quantum mechanics |
57R56 | Topological quantum field theories (aspects of differential topology) |
57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |