A bordered Chekanov-Eliashberg algebra

TL;DR

This paper introduces a new algebraic framework for Legendrian knots using bordered Chekanov-Eliashberg algebras, enabling decomposition and analysis of their invariants through van Kampen theorems and related tools.

Contribution

It develops a bordered algebraic approach for Legendrian knots, establishing van Kampen theorems and constructing maps for tangle replacements, advancing the algebraic understanding of Legendrian invariants.

Findings

Established a van Kampen theorem for Chekanov-Eliashberg algebras

Constructed maps between invariants under tangle replacements

Derived a Mayer-Vietoris sequence for linearized contact homology

Abstract

Given a front projection of a Legendrian knot $K$ in $\mathbb{R}^{3}$ which has been cut into several pieces along vertical lines, we assign a differential graded algebra to each piece and prove a van Kampen theorem describing the Chekanov-Eliashberg invariant of $K$ as a pushout of these algebras. We then use this theorem to construct maps between the invariants of Legendrian knots related by certain tangle replacements, and to describe the linearized contact homology of Legendrian Whitehead doubles. Other consequences include a Mayer-Vietoris sequence for linearized contact homology and a van Kampen theorem for the characteristic algebra of a Legendrian knot.