Department of Mathematics

Rutgers University

Piscataway, NJ 08854

edf41@rutgers.edu

Rutgers University

Piscataway, NJ 08854

edf41@rutgers.edu

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.

- ( 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

*H*.^{1}_{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.