{"id":{"repo_id":"cornell","oai_identifier":"oai:ecommons.cornell.edu:1813/64984"},"canonical_url":"https://search.dev.ndltd.org/etd/cornell/oai:ecommons.cornell.edu:1813/64984","repository":{"repo_id":"cornell","name":"Cornell University","base_url":"https://ecommons.cornell.edu/server/oai/request"},"display":{"title":"On conjectures related to character varieties of knots and Jones polynomials","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_html":"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_has_math":false,"creators":["Gallagher, Joseph"],"institution":"Cornell University","degree_name":"Ph. 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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."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["On conjectures related to character varieties of knots and Jones polynomials"]}]}],"canonical_facts":{"dc:contributor.committeemember":["Manning, Jason F.","Aguiar, Marcelo"],"dc:creator":["Gallagher, Joseph"],"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:format.mimetype":["application/pdf"],"dc:identifier.doi":["https://doi.org/10.7298/c8yt-3056"],"dc:identifier.other":["ProQuest Submission ID: 11148","ProQuest Publication ID: 10974051"],"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"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Doctor of Philosophy"],"thesis:degree_name":["Ph. D., Mathematics"],"thesis:institution_name":["Cornell University"]},"updated_at":"2026-07-24T01:49:10Z"}