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The Khovanov homology of knots

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thesis
posted on 15.11.2021, 13:48 by Le Gros, Giovanna

The Khovanov homology is a knot invariant which first appeared in Khovanov's original paper of 1999, titled ``a categorification of the Jones polynomial.'' This thesis aims to give an exposition of the Khovanov homology, including a complete background to the techniques used. We start with basic knot theory, including a definition of the Jones polynomial via the Kauffman bracket. Next, we cover some definitions and constructions in homological algebra which we use in the description of our title. Next we define the Khovanov homology in an analogous way to the Kauffman bracket, using only the algebraic techniques of the previous chapter, followed closely by a proof that the Khovanov homology is a knot invariant. After this, we prove an isomorphism of categories between TQFTs and Frobenius objects, which finally, in the last chapter, we put in the context of the Khovanov homology. After this application, we discuss some topological techniques in the context of the Khovanov homology.

History

Copyright Date

01/01/2015

Date of Award

01/01/2015

Publisher

Te Herenga Waka—Victoria University of Wellington

Rights License

Author Retains Copyright

Degree Discipline

Mathematics

Degree Grantor

Te Herenga Waka—Victoria University of Wellington

Degree Level

Masters

Degree Name

Master of Science

Victoria University of Wellington Unit

Centre for Applied Cross-Cultural Research

ANZSRC Type Of Activity code

1 PURE BASIC RESEARCH

Victoria University of Wellington Item Type

Awarded Research Masters Thesis

Language

en_NZ

Victoria University of Wellington School

School of Mathematics, Statistics and Operations Research

Advisors

Whittle, Geoff