Rutgers Math REU Project

Frontiers of Knot Theory

The goal of this project is to understand the current research in the subfield of low dimensional topology, Knot Theory.

Such as the relation between the knot invariants: A-polynomial and Augmentation polynomial.

Weekly Log

( 20 May ) Week 1

Met with our group to define the structure of our investigations. Read survey articles and textbooks on Knot Theory. Reviewed topics from Topology such as Fundamental Groups and Homology.

• "Knots, links, braids, and 3-manifolds" by Prasolov and Sosinsky
• "A Survey of Knot Theory" by Akio Kawauchi
• "Knots Knotes" by Justin Roberts [ pdf ]

( 27 May ) Week 2

Tynan Lazarus presented important theorems and fundamental ideas from Homotopy Theory relevant to the study of Knot invariants.

I presented the basic material of Knot theory. link to come

( 03 June ) Week 3

Alejandro Ginory introduced representation theory and presented background material for the A-polynomial.

( 10 June ) Week 4

Doug Schultz presented Knot Contact Homology and the Augmentation polynomial.

I presented 2-Bridge knots after which we computed the A-polynomials of 2-bridge knots.

( 17 June ) Week 5

Studied symplectic geometry and smooth manifolds.

( 24 June ) Week 6

Start of a two week conference in Montreal; Physics and Mathematics of Link Homology SMS 2013 Universite de Montreal.

( 01 July ) Week 7

I presented the trivial case of deRham Cohomology and a proof that H¹_dR( π ) = Hom( π, R).

( 08 July ) Week 8

Tynan and I are compiling a survey article(and/or webpage) on the A-polynomial and augmentation invariants with explicit calculations.