Computing Homology Invariants of Legendrian Knots

TL;DR

This paper explores the computation of homology invariants of Legendrian knots using combinatorial methods, revealing limitations and distinctions in augmentation homology groups related to generating families and normal rulings.

Contribution

It introduces a combinatorial approach to compute homology groups of augmentations and demonstrates new properties and distinctions in these invariants for Legendrian knots.

Findings

If the projection has 4 cusps, then |P(L)| ≤ 1.

Two augmentations from the same ruling may have non-isomorphic homology groups.

Abstract

The Chekanov-Eliashberg differential graded algebra of a Legendrian knot L is a rich source of Legendrian knot invariants, as is the theory of generating families. The set P(L) of homology groups of augmentations of the Chekanov-Eliashberg algebra is an invariant, as is a count of objects from the theory of generating families called graded normal rulings. This article gives two results demonstrating the usefulness of computing the homology group of an augmentation using a combinatorial interpretation of a generating family called a Morse complex sequence. First, we show that if the projection of L to the xz-plane has exactly 4 cusps, then |P(L)| is less than or equal to 1. Second, we show that two augmentations associated to the same graded normal ruling by the many-to-one map between augmentations and graded normal rulings defined by Ng and Sabloff need not have isomorphic homology groups.