{"id":{"repo_id":"cornell","oai_identifier":"oai:ecommons.cornell.edu:1813/31119"},"canonical_url":"https://search.dev.ndltd.org/etd/cornell/oai:ecommons.cornell.edu:1813/31119","repository":{"repo_id":"cornell","name":"Cornell University","base_url":"https://ecommons.cornell.edu/server/oai/request"},"display":{"title":"Kauffman Bracket Skein Modules And The Quantum Torus","abstract":"If M is a 3-manifold, the Kauffman bracket skein module is a vector space Kq (M ) functorially associated to M that depends on a parameter q ∈ C* . If F is a surface, then Kq (F x [0, 1]) is an algebra, and Kq (M ) is a module over Kq ((∂M ) x [0, 1]). One motivation for the definition is that if L [SUBSET OF] S 3 is a knot, then the (colored) Jones polynomials Jn (L) ∈ C[q ±1 ] can be computed from Kq (S 3 \\ L). It was shown in [14] that Kq (T 2 x [0, 1]) ~ AZ2 , the subalgebra of the quantum =q torus XY = q 2 Y X which is invariant under the involution X [RIGHTWARDS ARROW] X [-]1 , Y [RIGHTWARDS ARROW] Y [-]1 . Our starting point is the observation that the category of AZ2 -modules is equivalent q to the category of modules over a simpler algebra, the crossed product Aq Z2 . We write ML for the image of Kq (S 3 \\ L) under this equivalence. Theorem 5.2.1 gives a simple formula showing Jn (L) can be computed from ML , and Corollary 5.3.3 shows a recursion relation for Jn (L) can be computed from ML (if ML is f.g. over C[X ±1 ]). In Chapter 6 we give an explicit description of ML when L is the trefoil. Conjecture 4.3.4 conjectures the general structure of ML for torus knots. The algebra Aq Z2 is the t = 1 subfamily of the double affine Hecke algebra Hq,t of type A1 . In Chapter 8 we give a new skein-theoretic realization of the + + spherical subalgebra Hq,t , and we also give a construction associating an Hq,t - module ML (t) to each knot L. In Chapter 9 we construct algebraic deformations of the skein module ML to a family of modules ML (t) over Hq,t . In the case when L is the trefoil, we use these deformations to give example calculations of 2-variable polynomials Jn (q, t) that specialize to the colored Jones polynomials when t = 1.","abstract_html":"If M is a 3-manifold, the Kauffman bracket skein module is a vector space Kq (M ) functorially associated to M that depends on a parameter q ∈ C* . If F is a surface, then Kq (F x [0, 1]) is an algebra, and Kq (M ) is a module over Kq ((∂M ) x [0, 1]). One motivation for the definition is that if L [SUBSET OF] S 3 is a knot, then the (colored) Jones polynomials Jn (L) ∈ C[q ±1 ] can be computed from Kq (S 3 \\ L). It was shown in [14] that Kq (T 2 x [0, 1]) ~ AZ2 , the subalgebra of the quantum =q torus XY = q 2 Y X which is invariant under the involution X [RIGHTWARDS ARROW] X [-]1 , Y [RIGHTWARDS ARROW] Y [-]1 . Our starting point is the observation that the category of AZ2 -modules is equivalent q to the category of modules over a simpler algebra, the crossed product Aq Z2 . We write ML for the image of Kq (S 3 \\ L) under this equivalence. Theorem 5.2.1 gives a simple formula showing Jn (L) can be computed from ML , and Corollary 5.3.3 shows a recursion relation for Jn (L) can be computed from ML (if ML is f.g. over C[X ±1 ]). In Chapter 6 we give an explicit description of ML when L is the trefoil. Conjecture 4.3.4 conjectures the general structure of ML for torus knots. The algebra Aq Z2 is the t = 1 subfamily of the double affine Hecke algebra Hq,t of type A1 . In Chapter 8 we give a new skein-theoretic realization of the + + spherical subalgebra Hq,t , and we also give a construction associating an Hq,t - module ML (t) to each knot L. In Chapter 9 we construct algebraic deformations of the skein module ML to a family of modules ML (t) over Hq,t . In the case when L is the trefoil, we use these deformations to give example calculations of 2-variable polynomials Jn (q, t) that specialize to the colored Jones polynomials when t = 1.","abstract_has_math":false,"creators":["Samuelson, Peter"],"institution":"Cornell University","degree_name":"Ph. D., Mathematics","degree_level":"Doctor of Philosophy","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":["Knutson, Allen","Sjamaar, Reyer"],"year":2012,"date_issued":"2012-08-20","date_published":"2012-08-20","updated_at":"2026-07-24T01:49:08Z","subjects":["knot theory","quantum algebra"],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1813/31119","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Knutson, Allen","Sjamaar, Reyer"]},{"key":"dc:creator","label":"Author","values":["Samuelson, Peter"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2013-01-31T19:44:23Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-12-20T07:00:30Z"]},{"key":"dc:date.issued","label":"Date","values":["2012-08-20"]},{"key":"dc:type","label":"Dc Type","values":["dissertation or thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctor of Philosophy"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D., Mathematics"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Cornell University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["knot theory","quantum algebra"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1813/31119"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["If M is a 3-manifold, the Kauffman bracket skein module is a vector space Kq (M ) functorially associated to M that depends on a parameter q ∈ C* . If F is a surface, then Kq (F x [0, 1]) is an algebra, and Kq (M ) is a module over Kq ((∂M ) x [0, 1]). One motivation for the definition is that if L [SUBSET OF] S 3 is a knot, then the (colored) Jones polynomials Jn (L) ∈ C[q ±1 ] can be computed from Kq (S 3 \\ L). It was shown in [14] that Kq (T 2 x [0, 1]) ~ AZ2 , the subalgebra of the quantum =q torus XY = q 2 Y X which is invariant under the involution X [RIGHTWARDS ARROW] X [-]1 , Y [RIGHTWARDS ARROW] Y [-]1 . Our starting point is the observation that the category of AZ2 -modules is equivalent q to the category of modules over a simpler algebra, the crossed product Aq Z2 . We write ML for the image of Kq (S 3 \\ L) under this equivalence. Theorem 5.2.1 gives a simple formula showing Jn (L) can be computed from ML , and Corollary 5.3.3 shows a recursion relation for Jn (L) can be computed from ML (if ML is f.g. over C[X ±1 ]). In Chapter 6 we give an explicit description of ML when L is the trefoil. Conjecture 4.3.4 conjectures the general structure of ML for torus knots. The algebra Aq Z2 is the t = 1 subfamily of the double affine Hecke algebra Hq,t of type A1 . In Chapter 8 we give a new skein-theoretic realization of the + + spherical subalgebra Hq,t , and we also give a construction associating an Hq,t - module ML (t) to each knot L. In Chapter 9 we construct algebraic deformations of the skein module ML to a family of modules ML (t) over Hq,t . In the case when L is the trefoil, we use these deformations to give example calculations of 2-variable polynomials Jn (q, t) that specialize to the colored Jones polynomials when t = 1."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Kauffman Bracket Skein Modules And The Quantum Torus"]}]}],"canonical_facts":{"dc:contributor.committeemember":["Knutson, Allen","Sjamaar, Reyer"],"dc:creator":["Samuelson, Peter"],"dc:date.accessioned":["2013-01-31T19:44:23Z"],"dc:date.available":["2017-12-20T07:00:30Z"],"dc:date.issued":["2012-08-20"],"dc:description.abstract":["If M is a 3-manifold, the Kauffman bracket skein module is a vector space Kq (M ) functorially associated to M that depends on a parameter q ∈ C* . If F is a surface, then Kq (F x [0, 1]) is an algebra, and Kq (M ) is a module over Kq ((∂M ) x [0, 1]). One motivation for the definition is that if L [SUBSET OF] S 3 is a knot, then the (colored) Jones polynomials Jn (L) ∈ C[q ±1 ] can be computed from Kq (S 3 \\ L). It was shown in [14] that Kq (T 2 x [0, 1]) ~ AZ2 , the subalgebra of the quantum =q torus XY = q 2 Y X which is invariant under the involution X [RIGHTWARDS ARROW] X [-]1 , Y [RIGHTWARDS ARROW] Y [-]1 . Our starting point is the observation that the category of AZ2 -modules is equivalent q to the category of modules over a simpler algebra, the crossed product Aq Z2 . We write ML for the image of Kq (S 3 \\ L) under this equivalence. Theorem 5.2.1 gives a simple formula showing Jn (L) can be computed from ML , and Corollary 5.3.3 shows a recursion relation for Jn (L) can be computed from ML (if ML is f.g. over C[X ±1 ]). In Chapter 6 we give an explicit description of ML when L is the trefoil. Conjecture 4.3.4 conjectures the general structure of ML for torus knots. The algebra Aq Z2 is the t = 1 subfamily of the double affine Hecke algebra Hq,t of type A1 . In Chapter 8 we give a new skein-theoretic realization of the + + spherical subalgebra Hq,t , and we also give a construction associating an Hq,t - module ML (t) to each knot L. In Chapter 9 we construct algebraic deformations of the skein module ML to a family of modules ML (t) over Hq,t . In the case when L is the trefoil, we use these deformations to give example calculations of 2-variable polynomials Jn (q, t) that specialize to the colored Jones polynomials when t = 1."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/1813/31119"],"dc:language.iso":["en_US"],"dc:subject":["knot theory","quantum algebra"],"dc:title":["Kauffman Bracket Skein Modules And The Quantum Torus"],"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":"2026-07-24T01:49:08Z"}