Geometric Invariants in Contact Structures on 3-Manifolds

Citation

Zare, Douglas J. (1999) Geometric Invariants in Contact Structures on 3-Manifolds. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ywkx-tc69. https://resolver.caltech.edu/CaltechTHESIS:11212019-165707283

Abstract

Manifolds that have serious reasons to be odd-dimensional usually carry natural contact structures. — V. I. Arnold

Contact structures are important geometric structures on smooth, odd-dimensional manifolds. This thesis studies contact structures on 3-manifolds, where the theory is enriched by the study of knots and connections to the geometry of 4-manifolds.

In this thesis, we construct a geometric invariant of (framed) knots transverse to contact structures, size, and investigate its connections with Dehn surgeries, symplectic cobordisms, and geometric intersection invariants under contactomorphisms.

In chapter 2, we characterize tight and overtwisted contact structures in terms of the sizes of their transverse knots. We define surgeries of contact 3-manifolds along transverse curves, and give some upperbounds for sizes of knots after such surgeries. We show that certain reducing surgeries are precisely the results of certain symplectic cobordisms. Any transverse knot in a coorientable contact structure has such a family of such surgeries. This suggests that certain contact surgeries preserve tightness, and in section 3.2 we present some partial results in this direction. We provide some methods for obtaining lower bounds for the sizes of knots.

In chapter 3, we study the intersections of knots with surfaces under contactomorphisms. We study the local action of contactomorphisms on arcs near surfaces in contact 3-manifolds and the intersections of overtwisted discs with knots and overtwisted unknots with surfaces.

Thesis Files