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. https://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.)) | |||||||||
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| 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) | |||||||||
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| Defense Date: | 15 May 2015 | |||||||||
| Non-Caltech Author Email: | rfelliot (AT) gmail.com | |||||||||
| Record Number: | CaltechTHESIS:05292015-095300444 | |||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechTHESIS:05292015-095300444 | |||||||||
| DOI: | 10.7907/Z9ST7MRK | |||||||||
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| 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: | 05 Jul 2022 19:12 |
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