Geometry and Topology Seminar

This talk will revolve around classical braids considered from several perspectives. On the one hand, the braid group B_n is the mapping class group of a disk with n punctures, so that we can look at the dynamics of the braid monodromy and properties such as right-veering (roughly, twisting arcs to the right). Some of these properties are related to braid orderings, which can be viewed from a more algebraic perspective. Finally, from a contact topology viewpoint, a closed braid around the z-axis gives a link which is transverse to contact planes in the standard contact 3-space. Equivalence of braids up to certain stabilization corresponds to isotopy through transverse links. Transverse isotopy is quite subtle: first effective invariants to detect non-isotopic knots (beyond basic self-linking) were developed in 2006 by P.Ozsvath-Z.Szabo-D.Thurston by means of knot Floer homology. We will discuss the effect of (non)right-veering braid monodromy on the contact-topological properties of the corresponding transverse link and prove some non-vanishing results for the transverse invariants in knot Floer homology. Our proofs use braid orderings and the combinatorial approach to knot Floer homology; ingredients from all the different perspectives will be explained in the talk.