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Topological strings, double affine Hecke algebras, and exceptional knot homology

Citation

Elliot, Ross Filip (2015) Topological strings, double affine Hecke algebras, and exceptional knot homology. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9ST7MRK. http://resolver.caltech.edu/CaltechTHESIS:05292015-095300444

Abstract

In this thesis, we consider two main subjects: refined, composite invariants and exceptional knot homologies of torus knots. The main technical tools are double affine Hecke algebras ("DAHA") and various insights from topological string theory.

In particular, we define and study the composite DAHA-superpolynomials of torus knots, which depend on pairs of Young diagrams and generalize the composite HOMFLY-PT polynomials from the full HOMFLY-PT skein of the annulus. We also describe a rich structure of differentials that act on homological knot invariants for exceptional groups. These follow from the physics of BPS states and the adjacencies/spectra of singularities associated with Landau-Ginzburg potentials. At the end, we construct two DAHA-hyperpolynomials which are closely related to the Deligne-Gross exceptional series of root systems.

In addition to these main themes, we also provide new results connecting DAHA-Jones polynomials to quantum torus knot invariants for Cartan types A and D, as well as the first appearance of quantum E6 knot invariants in the literature.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Knot homology, DAHA, Topological string theory
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Apostol Award For Excellence In Teaching In Mathematics, 2012
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Gukov, Sergei
Thesis Committee:
  • Gukov, Sergei (chair)
  • Graber, Thomas B.
  • Marcolli, Matilde
  • Ni, Yi
Defense Date:15 May 2015
Non-Caltech Author Email:rfelliot (AT) gmail.com
Record Number:CaltechTHESIS:05292015-095300444
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:05292015-095300444
DOI:10.7907/Z9ST7MRK
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/1503.01441arXivRefined composite invariants of torus knots via DAHA
http://arxiv.org/abs/1505.01635arXivExceptional knot homology
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8920
Collection:CaltechTHESIS
Deposited By: Ross Elliot
Deposited On:02 Jun 2015 19:09
Last Modified:12 Apr 2016 22:12

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