On October 20th, the Geometry Group will host PATCH, our joint seminar with Bryn Mawr, Haverford and UPenn. The two speakers are Jason DeBlois from University of Pittsburgh and Matthew Hedden from Michigan State University.
DeBlois will speak at 3:40, on:
Title: Algebraic invariants, mutation, and commensurability of link complements
Abstract: I’ll describe a family of two-component links with the property that many algebraic invariants of their complements can be easily computed, and describe the commensurability relation among its members. Some mutants have commensurable complements and others do not. I’ll relate this to some open questions about knot complements.
Hedden will speak at 5:00, on:
Title: Contact structures associated to “rational” open books and their invariants
Abstract: A well-worn construction of Thurston and Winkelnkemper associates an essentially unique contact structure to an open book decomposition of a 3-manifold. Such a decomposition is essentially a choice of fibered knot or link in the 3-manifold, i.e. a link whose complement is a surface bundle over the circle in a “particular way”. I’ll discuss how to relax this “particular way” so knots which aren’t even null-homologous can still be considered fibered. The generalized open book structures that result are also related to contact geometry, and I’ll discuss invariants of these contact structures coming from Heegaard Floer homology. Our invariants can be fruitfully employed to populate the contact geometric universe with examples, and to better understand how it behaves under Dehn surgery. Using this latter understanding, I’ll discuss possible implications for the Berge Conjecture, a purely topological conjecture about the knots in the 3-sphere on which one can perform surgery and obtain lens spaces. This is joint work with Olga Plamenevskaya.