- Series
- Geometry Topology Seminar
- Time
- Monday, November 6, 2017 - 2:30pm for 2.5 hours
- Location
- Boyd 304
- Speaker
- Peter Lambert-Cole and Alex Zupan – Georgia Tech and Univ. Nebraska Lincoln
- Organizer
- Caitlin Leverson
Peter Lambert-Cole: Mutant knots are notoriously hard to distinguish. Many, but not all, knot invariants take the same value on mutant pairs. Khovanov homology with coefficients in Z/2Z is known to be mutation-invariant, while the bigraded knot Floer homology groups can distinguish mutants such as the famous Kinoshita-Terasaka and Conway pair. However, Baldwin and Levine conjectured that delta-graded knot Floer homology, a singly-graded reduction of the full invariant, is preserved by mutation. In this talk, I will give a new proof that Khovanov homology mod 2 is mutation-invariant. The same strategy can be applied to delta-graded knot Floer homology and proves the Baldwin-Levine conjecture for mutations on a large class of tangles. -----------------------------------------------------------------------------------------------------------------------------------------------Alex Zupan: Generally speaking, given a type of manifold decomposition, a natural
problem is to determine the structure of all decompositions for a fixed
manifold. In particular, it is interesting to understand the space of
decompositions for the simplest objects. For example, Waldhausen's
Theorem asserts that up to isotopy, the 3-sphere has a unique Heegaard
splitting in every genus, and Otal proved an analogous result for
classical bridge splittings of the unknot. In both cases, we say that
these decompositions are "standard," since they can be viewed as generic
modifications of a minimal splitting. In this talk, we examine a
similar question in dimension four, proving that -- unlike the situation
in dimension three -- the unknotted 2-sphere in the 4-sphere admits a
non-standard bridge trisection. This is joint work with Jeffrey Meier.