Generalized normal rulings and invariants of Legendrian solid torus links

Mikhail Lavrov and Dan Rutherford

Vol. 258 (2012), No. 2, 393–420

DOI: 10.2140/pjm.2012.258.393

Abstract

For Legendrian links in the 1-jet space of S¹ we show that the 1-graded ruling polynomial may be recovered from the Kauffman skein module. For such links a generalization of the notion of normal ruling is introduced. We show that the existence of such a generalized normal ruling is equivalent to sharpness of the Kauffman polynomial estimate for the Thurston–Bennequin number as well as to the existence of an ungraded augmentation of the Chekanov–Eliashberg DGA. Parallel results involving the HOMFLY-PT polynomial and 2-graded generalized normal rulings are established.

Keywords

Legendrian knot, Kauffman polynomial, skein module, normal ruling

Mathematical Subject Classification 2010

Primary: 57M27, 57R17

Milestones

Received: 4 October 2011

Revised: 8 December 2011

Accepted: 18 December 2011

Published: 1 August 2012