Intended Audience: Research mathematicians, professors of mathematics, graduate students in mathematics, and advanced undergraduate students in mathematics.
In this 1 hour episode, we see a presentation by Adam Saltz, a mathematician and graduate student at Boston College, on a new invariant of transverse knots in links coming from Khovanov homology.
In the talk, Adam discusses some of the details contained in his paper with Diana Hubbard, An annular refinement of the transverse element in Khovanov homology. Here is the abstract to their paper:
We construct a braid conjugacy class invariant κ by refining Plamenevskaya’s transverse element ψ in Khovanov homology via the annular grading. While κ is not an invariant of transverse links, it distinguishes some braids whose closures share the same classical invariants but are not transversely isotopic. Using κ we construct an obstruction to negative destabilization (stronger than ψ) and a solution to the word problem in braid groups. Also, κ is a lower bound on the length of the spectral sequence from annular Khovanov homology to Khovanov homology, and we obtain concrete examples in which this spectral sequence does not collapse immediately. In addition, we study these constructions in reduced Khovanov homology and illustrate that the two reduced versions are fundamentally different with respect to the annular filtration.
This video and paper are aimed at mathematicians, graduate students and undergraduates with lots of experience in topology. However, high school students who think they are potential math geniuses may still enjoy looking at it to see what advanced theorems and proofs look like.
As always, comments are welcome!
CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636 and NSF grant DMS-1249708