A cord algebra for tori in three-space

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 $LC{H}_0({\mathcal{L}}_{+}^{\ast }{T}_K)$ 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 $T_K$.

PhD thesis