A cord algebra for tori in three-space

TL;DR

This paper constructs a cord algebra model for tori around knots in three-space, relating it to Legendrian contact homology and using Morse theory and holomorphic curves.

Contribution

It introduces a Morse model for cord algebra of tori around knots, linking it to Legendrian contact homology via holomorphic curve techniques.

Findings

Identifies $Cord(T_K; \\mathbb{Z})$ with $Cord(K; \\mathbb{Z})$ using multiple time scale dynamics.

Connects cord algebra to 0-th degree Legendrian contact homology of the conormal bundle.

Provides a geometric interpretation via $J$-holomorphic curves with arboreal singularities.

Abstract

Given a thin torus $T_K$ around a knot $K\subset \mathbb{R}^3$, we construct Morse models of cord algebra $Cord(T_K)$ with $\mathbb{Z}$ and loop space coefficients. Using the Multiple time scale dynamics we identify $Cord(T_K; \mathbb{Z})$ with $Cord(K; \mathbb{Z})$. In combination with the works of Cieliebak-Ekholm-Latschev-Ng and Petrak this indirectly relates $Cord(T_K)$ to $0$-th degree Legendrian contact homology $LCH_0(\mathcal{L}^\ast_+ T_K)$ of one component of the unit conormal bundle over $T_K$. Our definition of $Cord(T_K)$ is motivated by $J$-holomorphic curves with boundary on the Lagrangian submanifold $L^\ast_+ T_K\cup\mathbb{R}^3$ with an arboreal singularity along the torus $T_K$.