Lenhard Ng

Computable Legendrian invariants

Abstract We establish tools to facilitate the computation and application of the Chekanov–Eliashberg differential graded algebra (DGA), a Legendrian-isotopy invariant of Legendrian knots and links in standard contact three space. More specifically, we reformulate the DGA in terms of front projections, and introduce the characteristic algebra, a new invariant derived from the DGA. We use our techniques to distinguish between several previously indistinguishable Legendrian knots and links.

Framed knot contact homology

We extend knot contact homology to a theory over the ring

A Legendrian Thurston-Bennequin bound from Khovanov homology

\mathbb{Z}[\lambda^{\pm 1},\mu^{\pm 1}]

Combinatorial knot contact homology and transverse knots

, with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in

Knot and braid invariants from contact homology II

S^3

Maximal Thurston–Bennequin number of two-bridge links

and can be generalized to knots in arbitrary manifolds, distinguishes the unknot and can distinguish mutants. It contains the Alexander polynomial and naturally produces a two-variable polynomial knot invariant which is related to the

Filtrations on the knot contact homology of transverse knots

A

Legendrian and transverse twist knots

-polynomial.

ON ARC INDEX AND MAXIMAL THURSTON-BENNEQUIN NUMBER

We establish an upper bound for the Thurston-Bennequin num- ber of a Legendrian link using the Khovanov homology of the underlying topological link. This bound is sharp in particular for all alternating links, and knots with nine or fewer crossings. AMS Classification 57M27; 57R17, 53D12

An Atlas of Legendrian Knots

Abstract We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and produces a three-variable knot polynomial related to the A -polynomial. We provide a number of computations of transverse homology that demonstrate its effectiveness in distinguishing transverse knots, including knots that cannot be distinguished by the Heegaard Floer transverse invariants or other previous invariants.