2026-09-20T08:44:26Zhttps://ecommons.cornell.edu/server/oai/requestoai:ecommons.cornell.edu:1813/649842026-05-15T19:51:56Zcom_1813_35col_1813_47
Gallagher, Joseph
Berest, Yuri
Manning, Jason F.
Aguiar, Marcelo
2019-04-02T14:01:18Z
2019-04-02T14:01:18Z
2018-12-30
ProQuest Submission ID: 11148
ProQuest Publication ID: 10974051
https://hdl.handle.net/1813/64984
https://doi.org/10.7298/c8yt-3056
10758124
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.
en_US
Attribution 4.0 International
https://creativecommons.org/licenses/by/4.0/
Mathematics
On conjectures related to character varieties of knots and Jones polynomials
dissertation or thesis
application/pdf
Mathematics
Cornell University
Doctor of Philosophy
Ph. D., Mathematics
https://hdl.handle.net/1813/59810
Publication
Cornell Theses and Dissertations
Gallagher, Joseph
5893a6ea-7af3-41d7-abc6-04bcd26ab5df
open.access
open.access
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<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.</Abstract>
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