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The Khovanov homotopy type and Conway mutation Garbuz, Benjamin
Abstract
Khovanov homology is a combinatorially-defined invariant of knots and links, with various generalizations to tangles. Recently, Lawson, Lipshitz, and Sarkar generalized Khovanov homology to a spectrum-valued Khovanov homotopy type, from which the Khovanov homology can be recovered. This thesis is primarily a ground-up survey of the Khovanov homotopy type; beginning with the Jones polynomial, we weave our way through Khovanov homology and the Khovanov homotopy type for links, before finishing with the construction of the Khovanov homotopy type for tangles. Throughout, we place a special emphasis on Conway mutation, an operation on links which involves replacing a tangle within a link by a related tangle. Despite its non-triviality, Conway mutation is impossible to detect with the Jones polynomial, and difficult to detect with Khovanov homology. The extent to which the Khovanov homotopy type is able to detect mutation is an open question, and the Khovanov homotopy type for tangles seems to be particularly well-suited for investigating this question.
Item Metadata
Title |
The Khovanov homotopy type and Conway mutation
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Creator | |
Supervisor | |
Publisher |
University of British Columbia
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Date Issued |
2021
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Description |
Khovanov homology is a combinatorially-defined invariant of knots and links, with various generalizations to tangles. Recently, Lawson, Lipshitz, and Sarkar generalized Khovanov homology to a spectrum-valued Khovanov homotopy type, from which the Khovanov homology can be recovered. This thesis is primarily a ground-up survey of the Khovanov homotopy type; beginning with the Jones polynomial, we weave our way through Khovanov homology and the Khovanov homotopy type for links, before finishing with the construction of the Khovanov homotopy type for tangles. Throughout, we place a special emphasis on Conway mutation, an operation on links which involves replacing a tangle within a link by a related tangle. Despite its non-triviality, Conway mutation is impossible to detect with the Jones polynomial, and difficult to detect with Khovanov homology. The extent to which the Khovanov homotopy type is able to detect mutation is an open question, and the Khovanov homotopy type for tangles seems to be particularly well-suited for investigating this question.
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Language |
eng
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Date Available |
2021-10-20
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0402561
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Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2021-11
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Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International