A Homological Approach to Relative Knot Invariants

TL;DR

This paper introduces relative invariants for Legendrian and transverse knots in contact 3-manifolds, extending classical invariants to knots homologous to a fixed reference, and establishes their well-definedness and basic properties.

Contribution

It develops a homological framework for defining and analyzing relative knot invariants, providing new tools for contact topology.

Findings

Relative invariants are well-defined for homologous knots.

Basic properties of these invariants are established.

The approach extends classical invariants to a broader context.

Abstract

We define relative versions of the classical invariants of Legendrian and transverse knots in contact 3-manifolds for knots that are homologous to a fixed reference knot. We show these invariants are well-defined and give some basic properties.