Reeb orbits that force topological entropy

We develop a forcing theory of topological entropy for Reeb flows in dimension three. A transverse link L in a closed contact $3$ -manifold $(Y,\xi )$ is said to force topological entropy if $(Y,\xi )$ admits a Reeb flow with vanishing topological entropy, and every Reeb flow on $(Y,\xi )$ realizing L as a set of periodic Reeb orbits has positive topological entropy. Our main results establish topological conditions on a transverse link L, which imply that L forces topological entropy. These conditions are formulated in terms of two Floer theoretical invariants: the cylindrical contact homology on the complement of transverse links introduced by Momin [A. Momin. J. Mod. Dyn.5 (2011), 409–472], and the strip Legendrian contact homology on the complement of transverse links, introduced by Alves [M. R. R. Alves. PhD Thesis, Université Libre de Bruxelles, 2014] and further developed here. We then use these results to show that on every closed contact $3$ -manifold that admits a Reeb flow with vanishing topological entropy, there exist transverse knots that force topological entropy.

Tight contact structures via admissible transverse surgery

We investigate the line between tight and overtwisted for surgeries on fibered transverse knots in contact 3-manifolds. When the contact structure [Formula: see text] is supported by the fibered knot [Formula: see text], we obtain a characterization of when negative surgeries result in a contact structure with nonvanishing Heegaard Floer contact class. To do this, we leverage information about the contact structure [Formula: see text] supported by the mirror knot [Formula: see text]. We derive several corollaries about the existence of tight contact structures, L-space knots outside [Formula: see text], nonplanar contact structures, and nonplanar Legendrian knots.

A CONSTRUCTION OF TRANSVERSELY NON-SIMPLE KNOT TYPES

For n = 1, 2 and 3, we construct a pair of transverse knots T1 and [Formula: see text], in the standard contact 3-sphere, satisfying the following properties: (1) the topological knot type of T1 is the same as that of [Formula: see text], (2) the self-linking number of T1 is equal to that of [Formula: see text], (3) [Formula: see text] is obtained from a transverse knot T2 by n stabilizations, and (4) T1 is not transversely isotopic to [Formula: see text].