Informal topology seminar

Time and location: Thursdays, 11am-12pm, room 622

About: This is a weekly seminar that gives participants the opportunity to present their own work or material that they are learning. The goal is to foster discussion, keep one another updated on our research, and create opportunities for collaboration. To receive seminar announcements, join our "informal-top" Google group.

Schedule:

  • February 26: Kyle Hayden, Exotica from knot traces, part I
  • March 5: Kyle Hayden, Exotica from knot traces, part II
  • March 14: Akram Alishahi, A braid invariant related to knot Floer homology and Khovanov homology
  • March 21: Spring break
  • March 28: No seminar
  • April 4: No seminar
  • April 11: Nathan Dowlin, A spectral sequence from Khovanov homology to knot Floer homology
  • April 18: Linh Truong, More concordance homomorphisms from knot Floer homology
  • April 25: Oleg Lazarev, Kirby calculus for symplectic manifolds
  • May 2: Melissa Zhang, Some annular concordance invariants from Khovanov homology

Details:

  • Exotica from knot traces, Kyle Hayden

A knot trace is a simple 4-manifold built by attaching a single 2-handle to the 4-ball along a knot in S^3. We'll discuss how knot traces play into the construction of exotic smooth structures on R^4 and "relatively" exotic structures on simple contractible 4-manifolds called Mazur manifolds. Then we'll describe new examples of exotic smooth structures on knot traces and use these to produce the first examples of ("absolutely") exotic smooth structures on Mazur manifolds. This new material is work in progress joint with Lisa Piccirillo.

  • A braid invariant related to knot Floer homology and Khovanov homology, Akram Alishahi

In this talk, I will sketch the definition of an algebraic braid invariant which is closely related to both Khovanov homology and the refinement of knot Floer homology into tangle invariants.

  • More concordance homomorphisms from knot Floer homology, Linh Truong

I will describe an infinite family of integer-valued concordance homomorphisms defined using knot Floer homology. I will discuss applications to topologically slice knots, concordance genus, and concordance unknotting number. This is joint work with Irving Dai, Jen Hom, and Matt Stoffregen.

  • Some annular concordance invariants from Khovanov homology, Melissa Zhang

A link in the complement of an unknot in S^3 is called "annular" because it lives in a thickened annulus and thus has a diagram on the annulus. Such annular settings are relevant to many fields of low-dimensional topology, including, for example, the study of transverse links in contact topology and the construction of satellite operators on the knot concordance group. In this talk, I'll describe some Upsilon-like invariants arising from filtrations on complexes related to Khovanov homology, focusing on my work in progress with Linh: we construct a 2-parameter family of annular concordance invariants by filtering Sarkar, Seed, and Szabó's perturbation of Szabó's geometric spectral sequence.