Mathematical Research Letters

Volume 24 (2017)

Number 6

The cardinality of the augmentation category of a Legendrian link

Pages: 1845 – 1874



Lenhard Ng (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Dan Rutherford (Department of Mathematical Sciences, Ball State University, Muncie, Indiana, U.S.A.)

Vivek Shende (Department of Mathematics, University of California at Berkeley)

Steven Sivek (Department of Mathematics, Imperial College London, United Kingdom)


We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact $\mathbb{R}^3$. 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.

Received 5 January 2016

Accepted 3 June 2016

Published 29 January 2018