Dissertation
Khovanov homotopy and its interaction with other link homology theories
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Summer 2023
DOI: 10.25820/etd.007211
Abstract
This thesis consists of three parts. The first part investigates the uniqueness of a construction by Sarkar-Seed-Szábo. They defined a combination theory that unifies two different link homology theories: Bar-Natan's purturbation and Szábo's geometric theory by introducing a collection of endomorphisms $\{h_i \}_{i=2}^{\infty}$. Their construction satisfies certain natural coherent properties. We show that the properties also led to the uniqueness of the collection of endomorphisms. In the second part we show an incompatibility between Szábo's geometric complex and (even) Khovanov homology. This is achieved by looking at the action of $sq_{1}$ on the geometric spectral sequence.
In the third part we construct an example of link homology theory using the spectrification of Lawson-Lipshitz-Sarkar. We show that for quasi-alternating links the associated Atiayh-Hirzebruch spectral seqeuence dies at the $E_2$ page.
Details
- Title: Subtitle
- Khovanov homotopy and its interaction with other link homology theories
- Creators
- Pravakar Paul
- Contributors
- Benjamin Cooper (Advisor)Keiko Kawamuro (Committee Member)Mohammad Tehrani (Committee Member)Isabel K Darcy (Committee Member)Ryan Kinser (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Summer 2023
- Publisher
- University of Iowa
- DOI
- 10.25820/etd.007211
- Number of pages
- ix, 65 pages
- Copyright
- Copyright 2023 Pravakar Paul
- Language
- English
- Date submitted
- 05/06/2023
- Description illustrations
- illustrations (some color)
- Description bibliographic
- Includes bibliographical references (pages 62-65).
- Public Abstract (ETD)
Knots and links have provided a deeper understanding of areas beyond mathematics, such as protein structure in Biology, statistical mechanics in Physics, and the structure of topological stereoisomers in Chemistry. Several mathematicians have constructed powerful link homology theories in order to gain a better understanding of low-dimensional topology, as well as to distinguish certain classes of knots and links. In this thesis we study interactions between different link homology theories. In the first part, we discover a uniqueness result on the combination of two certain link homology theories. In the second part, we demonstrate that two certain link homology theories are incompatible. Finally, using the space level description of Khovanov homology we construct a new link homology theory.
- Academic Unit
- Mathematics
- Record Identifier
- 9984454320302771
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