Abstract

  In the last three decades we have witnessed the growth of a fascinating mathematical theory called Knot Theory. Knot theory is a subeld of Topology, the study of properties of geometric objects that are preserved under deformations, and an advantage of knot theory over many other fields of mathematics is that much of the theory can be explained at an elementary level. The first part of the seminar is devoted to a short introduction to knot theory in a rapid penetration into key ideas and examples. We will present all essential background material of classical theory of knots, links and braids in S^3 culminating to the Jones polynomial, that we will define using the Kauffman bracket polynomial and via the Tempreley-Lieb algebra. We will then present a knot theoretic approach to the quantum group SL(2)_q via the Kauffman bracket polynomial. This way we also find a solution to the famous Yang-Baxter equation. We will then generalize the Jones polynomial to other 3-manifolds (skein modules). In particular, we will first introduce the notion of 3-manifolds, Dehn surgery, which is a simple way to obtain all 3-manifolds from S^3 and Kirby calculus, namely, an equivalence relation between (framed) knots that represent homeomorphic 3-manifolds. We will then generalize the Jones polynomial for knots and links in the Solid Torus, the Lens spaces L(p,q) and Homology Spheres. Our motivation is to construct 3-manifold invariants via quantum groups and our aim is to obtain a uniform algebraic approach to these (Witten) invariants, which remains an open problem in Low-Dimensional-Topology.

Useful Material (in English) for the seminar "Low-dimensional topology" @ NTUA

1. The HOMFLYPT skein module of the Lens spaces L(p, 1) via braids (PhD thesis)

2. Combinatorial construction of the HOMFLYPT polynomial (presentation)

Old Notes (in Greek) for the graduate course "Knot theory and Applications" @ NTUA

1. Knot theory and applications to Chemistry and Biology (Msc. thesis)

2. Knot theory and applications to Chemistry and Biology (presentation)

3. The Alexander Polynomial (lecture notes)

4. Knot theory: Lecture notes (pdf file)

References (books)

1. The Knot Book by C.C. Adams

2. Knots and Physics by L.H. Kauffman

3. Knots and Links by D. Rolfsen

4. Knots and Links by P. Cromwell

5. Knots by A. Sossinsky

6. When topology meets chemistry by E. Flapan

7. Braid groups by C. Kassel and V. Turaev

8. Quantum Invariants by T. Ohtsuki

9. Knots, Links, Braids and 3-manifolds by V. V. Prasolov and A. B. Sossinsky

10. Notes on Basic 3-manifold topology, by A. Hatcher

11. 3-manifods, by D. Calegari

References (papers)

1. On types of knotted curves, by J. W. Alexander & G. B. Briggs

2. Topological invariants of knots and links, by J. W. Alexander

3. A brief history of Knot theory, by Erin Colberg

4. Braids: A survey, by Joan Birman & Tara Brendle

5. New Invariants in the Theory of Knots, by Louis Kauffman 

6. Statistical Mechanics and the Jones Polynomial, by Louis Kauffman

7. The Jones polynomial, by V. F. R. Jones

8. Modeling DNA with Knot theory: An introduction, by J. Tompkins

9. Knot theory papers

10. Skein modules, by J. H. Przytycki

11. Knot Theory and Applications to 3-Manifolds, by E. Schlatter

12. A Calculus for framed Links in S^3, by R. Kirby

13. Rational surgery Calculus: Extension of Kirby's Theorem, by D. Rolfsen

14. Papers on arXiv