February 26, 2016
I will present some background on the classical Tait conjectures for
Josh Greene, Alternating links and definite surfaces
I will describe a characterization of alternating links in terms intrinsic to
the link exterior and use it to derive some properties of these links,
including algorithmic detection and new proofs of some of Tait's conjectures.
Lenny Ng, Using cotangent bundles and symplectic geometry to define knot invariants
Symplectic geometry has recently emerged as a key tool in the study of
low-dimensional topology. One approach, championed by Arnol'd, is to examine
the topology of a smooth manifold through the symplectic geometry of its
cotangent bundle, building on the familiar concept of phase space from
classical mechanics. I'll describe a way to use this approach, combined with
the modern theory of Legendrian contact homology (which I'll also introduce),
to construct a rather powerful invariant of knots called "knot contact homology".
Lenny Ng, Knot contact homology and string topology
Although knot contact homology has its origins in symplectic geometry and
holomorphic curves, it has a surprising relation to a more classical knot
invariant, the fundamental group of the knot complement. I'll discuss how one
can use string topology to make this relation a bit less surprising (in joint
work with Kai Cieliebak, Tobias Ekholm, and Janko Latschev), and apply this to
show that knot contact homology characterizes various types of knots.