Computing Homology Invariants of Legendrian Knots
Abstract
The ChekanovEliashberg 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 ChekanovEliashberg 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 xzplane 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 manytoone map between augmentations and graded normal rulings defined by Ng and Sabloff need not have isomorphic homology groups.
 Publication:

arXiv eprints
 Pub Date:
 June 2014
 arXiv:
 arXiv:1407.0069
 Bibcode:
 2014arXiv1407.0069C
 Keywords:

 Mathematics  Symplectic Geometry;
 Mathematics  Geometric Topology;
 57M27;
 57R17 (Primary);
 57M25 (Secondary)
 EPrint:
 18 pages, 11 figures