Braid Loops with infinite monodromy on the Legendrian contact DGA

TL;DR

This paper introduces new examples of Legendrian links with infinite monodromy action on their contact DGA, leading to the first known Legendrian links with infinitely many Lagrangian fillings, expanding understanding in symplectic topology.

Contribution

It provides the first examples of Legendrian links with infinite order monodromy action and constructs infinitely many Lagrangian fillings, including non-positive braid closures and specific torus links.

Findings

First examples of Legendrian links with infinite monodromy action.

Construction of Legendrian links with infinitely many Lagrangian fillings.

Floer-theoretic proof for infinite fillings of certain torus links.

Abstract

We present the first examples of elements in the fundamental group of the space of Legendrian links in the standard contact 3-sphere whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links that can be shown to admit infinitely many Lagrangian fillings by Floer-theoretic techniques. These families include the first known Legendrian links with infinitely many fillings that are not rainbow closures of positive braids, and the smallest Legendrian link with infinitely many fillings known to date. We discuss how to use our examples to construct other links with infinitely many fillings, in particular giving the first Floer-theoretic proof that Legendrian (n,m) torus links have infinitely many Lagrangian fillings, if n is greater than 3 and m greater than 6, or (n,m)=(4,4),(4,5). In addition, for any given higher genus, we construct a Weinstein 4-manifold homotopic to the 2-sphere whose wrapped Fukaya category can distinguish infinitely many exact closed Lagrangian surfaces of that genus. A key technical ingredient behind our results is a new combinatorial formula for decomposable cobordism maps between Legendrian contact DGAs with integer (group ring) coefficients.