Ruling polynomials and augmentations over finite fields. (30th July 2014)

• Record Type: Journal Article
• Authors: Henry, Michael B.; Rutherford, Dan
• Is Part Of: Journal of topology. Volume 8:Part 1(2015)
• Page Start: 1
• Page End: 37
• Publication Date: 2014-07-30
• Subjects: Topology -- Periodicals; 514.05
• DOI: 10.1112/jtopol/jtu013
• Languages: English
• ISSNs: 1753-8416

Abstract

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) ofL into a finite field of orderq where the parameterm is a divisor of twice the rotation number ofL . 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 caseq = 2, we show the augmentation numbers, Aug m ( L, q ), are determined by specializing them ‐graded ruling polynomial, R L m ( z ), atz = q 1 / 2 ‐ q ‐ 1 / 2 . As a corollary, we deduce that the ruling polynomials are determined by the Legendrian contact homology DGA.