A hamiltonian graph G of order n is k-ordered, 2 â‰¤ k â‰¤ n, if for every sequence v1, v2, . . . , vk of k distinct vertices of G, there exists a hamiltonian cycle that encounters v1, v2, . . . , vk inâ€¦ (More)

We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations. This interpretation allows us to extend the knot invariant toâ€¦ (More)

We establish an upper bound for the Thurstonâ€“Bennequin number of a Legendrian link using the Khovanov homology of the underlying topological link. This bound is sharp in particular for allâ€¦ (More)

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 isâ€¦ (More)

The string-theoretic approach originated in a conjecture of â€™t Hooft relating largeâ€“N gauge theories and open string theories, and a subsequent observation by Witten [23] connecting Chernâ€“Simonsâ€¦ (More)

We present an atlas of Legendrian knots in standard contact three-space. This gives a conjectural Legendrian classification for all knots with arc index at most 9, including alternating knots throughâ€¦ (More)

We extend knot contact homology to a theory over the ring Z[Î»Â±1, Î¼Â±1], with the invariant given topologically and combinatorially. The improved invariant, which is defined for framed knots in S andâ€¦ (More)

We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology,â€¦ (More)

We write a program in Java to generate all grid diagrams of up to size 10. Using equivalence between grid diagrams modulo a set of Cromwell moves and classification of Legendrian links up toâ€¦ (More)