Structures and Computations in Annular Khovanov Homology
| dc.contributor.advisor | Lipshitz, Robert | |
| dc.contributor.author | Davis, Champ | |
| dc.date.accessioned | 2024-01-09T22:53:41Z | |
| dc.date.available | 2024-01-09T22:53:41Z | |
| dc.date.issued | 2024-01-09 | |
| dc.description.abstract | Let $L$ be a link in a thickened annulus. In [GLW17], Grigsby-Licata-Wehrli showed that the annular Khovanov homology of $L$ is equipped with an action of $\exsltwo$, the exterior current algebra of the Lie algebra $\sltwo$. In this dissertation, we upgrade this result to the setting of $L_\infty$-algebras and modules. That is, we show that $\exsltwo$ is an $L_\infty$-algebra and that the annular Khovanov homology of $L$ is an $L_\infty$-module over $\exsltwo$. Up to $L_\infty$-quasi-isomorphism, this structure is invariant under Reidemeister moves. In proving the above result, we include explicit formulas to compute the higher $L_\infty$-operations. Additionally, given a morphism $I: L' \to L$ of $L_\infty$-algebras, we define a restriction of scalars operation in the setting of $L_\infty$-modules and prove that it defines a functor $I^*: L-mod \to L'-mod$. A more abstract approach to this problem was recently given by Kraft-Schnitzer. Finally, computer code was written to aid in the study of the above $L_\infty$-module structure. We discuss various patterns that emerged from these computations, most notably one relating the torsion in the annular Khovanov homology groups and the location of the inner boundary of the annulus. | en_US |
| dc.identifier.uri | https://hdl.handle.net/1794/29197 | |
| dc.language.iso | en_US | |
| dc.publisher | University of Oregon | |
| dc.rights | All Rights Reserved. | |
| dc.subject | Annular Khovanov Homology | en_US |
| dc.subject | Knot Theory | en_US |
| dc.subject | L-infinity algebras | en_US |
| dc.title | Structures and Computations in Annular Khovanov Homology | |
| dc.type | Electronic Thesis or Dissertation | |
| thesis.degree.discipline | Department of Mathematics | |
| thesis.degree.grantor | University of Oregon | |
| thesis.degree.level | doctoral | |
| thesis.degree.name | Ph.D. |
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