Ruling polynomials and augmentations over finite fields

Abstract

For any Legendrian link, L, in ( R 3 , ker ( d z - y d x ) ) , we define invariants, Aug m ( L , q ) , as normalized counts of augmentations from the Legendrian contact homology differential graded algebra (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 from [Ng and Sabloff, ‘The correspondence between augmentations and rulings for Legendrian knots’, Pacific J. Math. 224 (2006) 141–150] for the case q = 2 , we show the augmentation numbers, Aug m ( L , q ) , are determined by specializing the m-graded ruling polynomial, R L m ( 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.