Knot contact homology, string topology, and the cord algebra

Abstract

The conormal Lagrangian $L_K$ of a knot $K$ in ${\mathbb{R}}^3$ is the submanifold of the cotangent bundle $T^{\ast }{\mathbb{R}}^3$ consisting of covectors along $K$ that annihilate tangent vectors to $K$. By intersecting with the unit cotangent bundle $S^{\ast }{\mathbb{R}}^3$, one obtains the unit conormal ${\Lambda }_K$, and the Legendrian contact homology of ${\Lambda }_K$ is a knot invariant of $K$, known as knot contact homology. We define a version of string topology for strings in ${\mathbb{R}}^3\cup L_K$ and prove that this is isomorphic in degree 0 to knot contact homology. The string topology perspective gives a topological derivation of the cord algebra (also isomorphic to degree 0 knot contact homology) and relates it to the knot group. Together with the isomorphism this gives a new proof that knot contact homology detects the unknot. Our techniques involve a detailed analysis of certain moduli spaces of holomorphic disks in $T^{\ast }{\mathbb{R}}^3$ with boundary on ${\mathbb{R}}^3\cup L_K$.

116 pages, v3: small changes, to appear in the Journal de l'École polytechnique - Mathématiques

References in corpus (1)

• Transverse string topology and the cord algebra

Cited by in corpus (1)

• Relative Calabi-Yau completions