Some variations of Khovanov homology for null homologous links in RP^3
Abstract/Contents
- Abstract
- Asaeda, Przytycki and Sikora extended Khovanov homology to links in $I$-bundles over surface. In particular, their construction works for links in the real projective space $\mathbb{RP}^3$. In this thesis, we will present two variations of their construction for null homologous links in $\mathbb{RP}^3$. First, we introduce Khovanov-type homologies which depend on an extra algebraic input $\alpha = (V_0, V_1, f, g)$, consisting of two graded vectors spaces and two maps between them. With some specific choice of $\alpha = \alpha_{\mathit{APS}}$, we recover a reduced version of the Asaeda-Przytycki-Sikora construction. With another choice of $\alpha = \alpha_{\mathit{HF}}$, we construct a spectral sequence from our theory converging to the Heegaard Floer homology of the even branched double cover of $\mathbb{RP}^3$, extending the usual Ozsv\'{a}th-Szab\'{o} spectral sequence for links in $S^3$. Second, we introduce Bar-Natan homology for null homologous links in $\mathbb{RP}^3$ over the field of two elements. It is a deformation of the Asaeda-Przytycki-Sikora construction, in the sense that the associated graded complex with respect to the quantum grading gives their chain complex. We also define an $s$-invariant from this deformation following the same recipe as for links in $S^3$, and establish certain genus bounds based on its properties.
Description
| Type of resource | text |
|---|---|
| Form | electronic resource; remote; computer; online resource |
| Extent | 1 online resource. |
| Place | California |
| Place | [Stanford, California] |
| Publisher | [Stanford University] |
| Copyright date | 2023; ©2023 |
| Publication date | 2023; 2023 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Author | Chen, Daren |
|---|---|
| Degree supervisor | Manolescu, Ciprian, 1978- |
| Thesis advisor | Manolescu, Ciprian, 1978- |
| Thesis advisor | Eliashberg, Y, 1946- |
| Thesis advisor | Li, Zhenkun |
| Degree committee member | Eliashberg, Y, 1946- |
| Degree committee member | Li, Zhenkun |
| Associated with | Stanford University, School of Humanities and Sciences |
| Associated with | Stanford University, Department of Mathematics |
Subjects
| Genre | Theses |
|---|---|
| Genre | Text |
Bibliographic information
| Statement of responsibility | Daren Chen. |
|---|---|
| Note | Submitted to the Department of Mathematics. |
| Thesis | Thesis Ph.D. Stanford University 2023. |
| Location | https://purl.stanford.edu/zh423zx6819 |
Access conditions
- Copyright
- © 2023 by Daren Chen
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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