The Trieste look at Knot Theory

TL;DR

This paper provides an historical overview of knot theory, explores Fox colorings and their relation to the Jones polynomial, and introduces a novel symplectic structure on tangles linking them to Lagrangian subspaces.

Contribution

It introduces a new symplectic structure on tangle boundaries, connecting knot theory to symplectic geometry and Lagrangian subspaces, with unpublished proof details.

Findings

Relation between 3-colorings and Jones polynomial

Development of symplectic structure for tangles

Tangles as Lagrangian submanifolds

Abstract

This paper is base on talks which I gave in May, 2010 at Workshop in Trieste (ICTP). In the first part we present an introduction to knots and knot theory from an historical perspective, starting from Summerian knots and ending on Fox 3-coloring. We show also a relation between 3-colorings and the Jones polynomial. In the second part we develop the general theory of Fox colorings and show how to associate a symplectic structure to a tangle boundary so that tangles becomes Lagrangians (a proof of this result has not been published before). Chapter VI of the book "KNOTS: From combinatorics of knot diagrams to combinatorial topology based on knots will be based on this paper.