The cardinality of the augmentation category of a Legendrian link

TL;DR

This paper introduces a new invariant called homotopy cardinality for the augmentation category of Legendrian links, linking it to the ruling polynomial and applying it to doubly Lagrangian slice knots.

Contribution

It defines the homotopy cardinality of the augmentation category and establishes its relation to the ruling polynomial, providing new insights into Legendrian link invariants.

Findings

Homotopy cardinality is an invariant of the augmentation category.

The homotopy cardinality is determined by the ruling polynomial.

Application to doubly Lagrangian slice knots demonstrates the invariant's utility.

Abstract

We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact R^3. This `homotopy cardinality' is an invariant of the category and allows for a weighted count of augmentations, which we prove to be determined by the ruling polynomial of the link. We present an application to the augmentation category of doubly Lagrangian slice knots.