Ruling polynomials and augmentations over finite fields

TL;DR

This paper introduces invariants for Legendrian links based on counts of augmentations over finite fields, linking these invariants to ruling polynomials and Legendrian contact homology.

Contribution

It generalizes previous results by relating augmentation counts over finite fields to ruling polynomials for all q, not just q=2.

Findings

Augmentation numbers are determined by ruling polynomials at specific values.

Ruling polynomials are fully determined by the Legendrian contact homology DGA.

Extension of Ng and Sabloff's result to arbitrary finite fields.

Abstract

For any Legendrian link, L, in (\R^3, \ker(dz-y\,dx)) we define invariants, Aug_m(L,q), as normalized counts of augmentations from the Legendrian contact homology DGA of L into a finite field of order q where the parameter m is a divisor of twice the rotation number of L. Generalizing a result of Ng and Sabloff for the case q =2, we show the augmentation numbers, Aug_m(L,q), are determined by specializing the m-graded ruling polynomial, R^m_L(z), at z = q^{1/2}-q^{-1/2}. As a corollary, we deduce that the ruling polynomials are determined by the Legendrian contact homology DGA.