On conjectures related to character varieties of knots and Jones polynomials
| dc.contributor.author | Gallagher, Joseph | |
| dc.contributor.chair | Berest, Yuri | |
| dc.contributor.committeeMember | Manning, Jason F. | |
| dc.contributor.committeeMember | Aguiar, Marcelo | |
| dc.date.accessioned | 2019-04-02T14:01:18Z | |
| dc.date.available | 2019-04-02T14:01:18Z | |
| dc.date.issued | 2018-12-30 | |
| dc.description.abstract | It is well known that the Kauffman Bracket Skein Module of a knot complement K_q(S^3 \ K) is canonically a module over the Z_2-invariants of the quantum torus, A_q^{Z_2}, and this module determines the colored Jones polynomials J_n(K; q) of the knot K. Berest and Samuelson identified a conjecture for knots under which a close variant of K_q(S^3 \ K) canonically becomes a module over a certain Double Affine Hecke Algebra, from which they defined a family of polynomials J_n(K; q; t_1; t_2) generalizing the classical polynomials of Jones. In this thesis an analogue of Habiro’s cyclotomic equation for the J_n(K; q) is discovered for J_n(K; q; t_1; t_2). An integrality result for the coefficients in this equation is found as a corollary, offering evidence for the conjecture of Berest and Samuelson for all knots. Separately, the conjecture of Berest and Samuelson is studied at the particular value q = -1 where it is known to relate to properties of SL_2(C)-character varieties of knots. Computational methods are used to establish that the conjecture holds for some non-invertible knots, which was not previously known. | |
| dc.identifier.doi | https://doi.org/10.7298/c8yt-3056 | |
| dc.identifier.other | Gallagher_cornellgrad_0058F_11148 | |
| dc.identifier.other | http://dissertations.umi.com/cornellgrad:11148 | |
| dc.identifier.other | bibid: 10758124 | |
| dc.identifier.uri | https://hdl.handle.net/1813/64984 | |
| dc.language.iso | en_US | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.subject | Mathematics | |
| dc.title | On conjectures related to character varieties of knots and Jones polynomials | |
| dc.type | dissertation or thesis | |
| dcterms.license | https://hdl.handle.net/1813/59810 | |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | Cornell University | |
| thesis.degree.level | Doctor of Philosophy | |
| thesis.degree.name | Ph. D., Mathematics |
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