Abstract: Assuming only basic algebra, present the necessary basic knot theory, tensor products, and homological algebra to then define and explore Khovanov homology (through explicit and explained example computations), and compute the homolgy of the (2,n) torus link.
Poster presentation of 12/15/2005 is here.
Full thesis "semi-final" draft of 04/22/2005 is here.
Paper status: "Finished", at a rather hefty 40 pages, considering I feel like I didn't finish what I wanted to.
Outline (04/23/2005):
1 Introduction
2 Basic Knot Theory
2.1 What is a knot?
2.2 Knot equivalence and the Reidemeister moves
2.3 A few more basic definitions...
3 The Jones Polynomial
3.1 The Kauffman bracket
3.2 The Jones polynomial
3.3 Skeins
3.4 Invariance of the Jones polynomial
3.5 Mirrors and the Jones polynomial
3.6 Why more?
4 Some Necessary Tools
4.1 Tensor products and duals
4.2 Graded vector spaces
4.3 Chain complexes
4.4 Homology
4.5 The graded Euler characteristic
5 Khovanov Homology
5.1 Vector spaces
5.2 Differentials
5.3 Invariants
5.4 Example computation of Kh(L)
5.5 Mirrors and the Khovanov homology
5.6 Invariance, proof of
5.6.1 R1
5.6.2 R1
5.6.3 R1
6 Some Computations
6.1 (2,n) torus links
References