Remarks on Legendrian Self-Linking

TL;DR

This paper introduces a new analytic approach to Legendrian self-linking in Euclidean space, generalizing the Thurston-Bennequin invariant through a reformulated Gauss linking integral inspired by gauge theory.

Contribution

It presents a novel analytic definition of Legendrian self-linking that extends the Thurston-Bennequin invariant using ideas from supersymmetric gauge theory.

Findings

Reformulation of the Gauss linking integral for Legendrian knots

Recovery of the Thurston-Bennequin invariant as a special case

Introduction of a new analytic invariant for Legendrian knots

Abstract

The Thurston-Bennequin invariant provides one notion of self-linking for any homologically-trivial Legendrian curve in a contact three-manifold. Here we discuss related analytic notions of self-linking for Legendrian knots in Euclidean space. Our definition is based upon a reformulation of the elementary Gauss linking integral and is motivated by ideas from supersymmetric gauge theory. We recover the Thurston-Bennequin invariant as a special case.