Braids, knots and contact structures

TL;DR

This paper reviews the relationships between braids, knots, and contact structures in 3-space, highlighting recent work that distinguishes certain transversal knot types sharing topological and invariant properties.

Contribution

It presents new results demonstrating the existence of distinct transversal knots with identical topological types and Bennequin invariants.

Findings

Existence of different transversal knot types with same topological knot and Bennequin invariant

Connections established between braids, knots, and contact geometry

Recent proofs distinguishing transversal knots in contact 3-space

Abstract

These notes were prepared to supplement the talk that I gave on Feb 19, 2004, at the First East Asian School of Knots and Related Topics, Seoul, South Korea. In this article I review aspects of the interconnections between braids, knots and contact structures on Euclidean 3-space. I discuss my recent work with William Menasco (arXiv math.GT/0310279)} and (arXiv math.GT/0310280). In the latter we prove that there are distinct transversal knot types in contact 3-space having the same topological knot type and the same Bennequin invariant.