Obstructing Lagrangian concordance for closures of 3-braids

TL;DR

This paper establishes obstructions to Lagrangian concordance between certain knots and the Legendrian unknot, using symplectic topology and contact homology, leading to new insights on contact manifold embeddings.

Contribution

It introduces a novel obstruction method for Lagrangian concordance using Weinstein handlebody diagrams and Legendrian contact homology, applicable to closures of 3-braids.

Findings

Knots as closures of 3-braids cannot be Lagrangian concordant to the Legendrian unknot unless they are the unknot.

Derived a contradiction by computing symplectic homology of symplectic fillings.

Identified an infinite family of contact manifolds that do not embed as contact type hypersurfaces in ^4.

Abstract

We show that any knot which is smoothly the closure of a 3-braid cannot be Lagrangian concordant to and from the maximum Thurston-Bennequin Legendrian unknot except the unknot itself. Our obstruction comes from drawing the Weinstein handlebody diagrams of particular symplectic fillings of cyclic branched double covers of knots in $S^3$. We use the Legendrian contact homology differential graded algebra of the links in these diagrams to compute the symplectic homology of these fillings to derive a contradiction. As a corollary, we find an infinite family of contact manifolds which are rational homology spheres but do not embed in $\mathbb{R}^4$ as contact type hypersurfaces.