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Lectures on Knot Homology and Quantum Curves

Gukov, Sergei and Saberi, Ingmar (2016) Lectures on Knot Homology and Quantum Curves. In: Physics and Mathematics of Link Homology. Contemporary Mathematics. No.680. American Mathematical Society , Providence, RI, pp. 59-97. ISBN 978-1-4704-1459-7. https://resolver.caltech.edu/CaltechAUTHORS:20160503-081755071

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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.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1090/conm/680/13700 DOITable of Contents
http://www.ams.org/books/conm/680/PublisherTable of Contents
http://arxiv.org/abs/1211.6075arXivDiscussion Paper
ORCID:
AuthorORCID
Gukov, Sergei0000-0002-9486-1762
Additional Information:© 2017 American Mathematical Society. This survey was presented in a series of lectures first at University of Notre Dame (2012 summer school on Topology and Field Theories at the Center for Mathematics), then at Stanford (summer school on Holomorphic curves and low-dimensional topology), and in other versions in Hamburg, Oberwolfach, Lisbon, and Heidelberg.
Subject Keywords:Knot invariants; quantization; categorification; Chern-Simons theory
Series Name:Contemporary Mathematics
Issue or Number:680
Record Number:CaltechAUTHORS:20160503-081755071
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160503-081755071
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:66603
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:03 May 2016 19:55
Last Modified:11 Feb 2020 19:00

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