Graduate Topology Seminar -- Spring 2006 -- Knot Theory

Meeting Time/Place: Tuesdays 2pm, LOV 104

1/31/06: Knot diagrams, Reidemeister moves
Handout 1: List of Topics
Homework 1
2/7/06: Colorability, Quandles and other invariants
Using Reidemeister's Theorem and colorability we prove that the trefoil is not ambient isotopic to the unknot. We also discuss properties of an extension of colorability called quandles.
2/14/06: Keis and Quandles
We show colorings correspond to homomorphisms of Knot Quandles to finite Keis. We also discuss fundamental group, Alexander polynomials, and Kauffman's generalized Knot Quandles.
2/21/06: Bracket polynomial, writhe and Jones polynomial
2/28/06: no seminar
3/14/06: Vassiliev Invariants I
We use the Jones polynomial to create an example of a Vassiliev Invariant of type i.
3/21/06: Vassiliev Invariants II
We define matrix diagrams and show that matrix Lie algebras give rise to weighted chord diagrams. Due to a theorem of Kontsevich and Bar-Natan, these in turn give top level evaluations for at least one Vassiliev invariant of type i, where i is the dimension of the Lie algebra.