An Introduction to Khovanov Homology

TL;DR

This paper introduces Khovanov homology, exploring its foundations through state summation, cobordism, and quantum models, linking it to the Jones polynomial and Hilbert space representations.

Contribution

It presents a comprehensive overview of Khovanov homology, including new perspectives via simplicial and quantum models, connecting algebraic and topological aspects.

Findings

Khovanov homology can be derived from the Kauffman bracket.

A quantum model links the Jones polynomial to Hilbert space traces.

The paper emphasizes the cobordism and simplicial approaches.

Abstract

This paper is an introduction to Khovanov homology, starting with the Kauffman bracket state summation, emphasizing the Bar-Natan Canopoloy and tangle cobordism approach. The paper discusses a simplicial approach to Khovanov homology and a quantum model for it so that the graded Euler characteristic that produces the Jones polynomial from Khovanov homology becomes the trace of a unitary transformation on a Hilbert space associated with the Khovanov Homology.