Cocycle knot invariants from quandle modules and generalized quandle homology

Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations...

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Autores principales: Carter, J.S., Elhamdadi, M., Graña, M., Saito, M.
Formato: JOUR
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00306126_v42_n3_p499_Carter
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_00306126_v42_n3_p499_Carter2023-10-03T14:40:44Z Cocycle knot invariants from quandle modules and generalized quandle homology Carter, J.S. Elhamdadi, M. Graña, M. Saito, M. Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander modules for classical knots. Second, 2-cocycles valued in non-abelian groups are used in a way similar to Hopf algebra invariants of classical knots. These invariants are shown to be of quantum type. Third, cocycles with group actions on coefficient groups are used to define quandle cocycle invariants for both classical knots and knotted surfaces. Concrete computational methods are provided and used to prove non-invertibility for a large family of knotted surfaces. In the classical case, the invariant can detect the chirality of 3-colorable knots in a number of cases. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00306126_v42_n3_p499_Carter
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
description Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander modules for classical knots. Second, 2-cocycles valued in non-abelian groups are used in a way similar to Hopf algebra invariants of classical knots. These invariants are shown to be of quantum type. Third, cocycles with group actions on coefficient groups are used to define quandle cocycle invariants for both classical knots and knotted surfaces. Concrete computational methods are provided and used to prove non-invertibility for a large family of knotted surfaces. In the classical case, the invariant can detect the chirality of 3-colorable knots in a number of cases.
format JOUR
author Carter, J.S.
Elhamdadi, M.
Graña, M.
Saito, M.
spellingShingle Carter, J.S.
Elhamdadi, M.
Graña, M.
Saito, M.
Cocycle knot invariants from quandle modules and generalized quandle homology
author_facet Carter, J.S.
Elhamdadi, M.
Graña, M.
Saito, M.
author_sort Carter, J.S.
title Cocycle knot invariants from quandle modules and generalized quandle homology
title_short Cocycle knot invariants from quandle modules and generalized quandle homology
title_full Cocycle knot invariants from quandle modules and generalized quandle homology
title_fullStr Cocycle knot invariants from quandle modules and generalized quandle homology
title_full_unstemmed Cocycle knot invariants from quandle modules and generalized quandle homology
title_sort cocycle knot invariants from quandle modules and generalized quandle homology
url http://hdl.handle.net/20.500.12110/paper_00306126_v42_n3_p499_Carter
work_keys_str_mv AT carterjs cocycleknotinvariantsfromquandlemodulesandgeneralizedquandlehomology
AT elhamdadim cocycleknotinvariantsfromquandlemodulesandgeneralizedquandlehomology
AT granam cocycleknotinvariantsfromquandlemodulesandgeneralizedquandlehomology
AT saitom cocycleknotinvariantsfromquandlemodulesandgeneralizedquandlehomology
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