Rational Symplectic Field Theory for Legendrian knots

TL;DR

This paper introduces a combinatorial invariant for Legendrian knots in contact three-space, extending contact homology by incorporating rational Symplectic Field Theory and counting holomorphic disks with multiple punctures.

Contribution

It develops a new combinatorial invariant that encodes rational SFT and extends contact homology for Legendrian knots, using string topology techniques.

Findings

Invariant captures Legendrian knot properties

Extends contact homology with rational SFT features

Counts holomorphic disks with multiple positive punctures

Abstract

We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary number of positive punctures. The construction uses ideas from string topology.