Dissertation: Topics in Khovanov Homology
Abstract: In this dissertation we work with Khovanov homology and its variants. Khovanov homology is a "categorification" of the Jones polynomial. It consists of graded chain complexes which, up to chain homotopy, are link invariants, and whose graded Euler characteristic is equal to the Jones polynomial of the link. Rasmussen's invariant gives a bound on the smooth 4-ball genus of a knot. We construct bounds on Rasmussen's invariant which are easily computable from any diagram. Using this construction, one also obtains representatives for the homology classes of the Lee homology of the knot. These bounds are sharp precisely for homogeneous knots, a class of knots containing both alternating knots and positive knots. We prove that a class of braid-positive links, more general than torus links, are Khovanov thick. From this observation we get infinite families of prime knots all of which are Khovanov thick. This is further evidence toward Khovanov's conjecture that all braid positive knots other than the T(2,2k + 1) torus knots are thick. We provide a technique for creating cobordisms between knots which result in injections at the homological level. The technique is applied and studied in the case of ribbon knots.Topological Data Analysis:
Persistent homology is the study of filtered chain complexes using algebraic tools. Filtered chain complexes can be created in a variety of settings. In work with David Meyer, I studied the persistent homology of random samples from Lorentzian manifolds. We use the induced causal set on the sampled points to recover topological information about the space from which it was sampled.Conferences Attended:
2010 MSRI Introductory Workshop: Homology Theories of Knots and Links2009 International Conference on Geometry and Topology, Athens, Georgia
2009 DARPA Topological Data Analysis Santa Barbara
2008 AMS Joint Meetings San Deigo CA
Previous Talks (at UCSD unless otherwise stated):
Advancement to CandidacyTopology Seminar: Rasmussen's proof of Milnor's conjecture on the unknotting number of torus knots
Quantum Seminar: Poset Dimensions
Quantum Seminar: A B_n Persistent Object
Quantum Seminar: Finite Topological Spaces
Quantum Seminar: Zigzag Persistence
Topology Seminar: Circle-Valued Morse Theory
Topology Seminar: Braided Monoidal Categories and Knot Theory
Quantum Seminar: Multidimensional Persistence
Topology Seminar: nCob and TQFTs
Quantum Seminar: Topological Data Analysis
Topology Seminar: Rational Homotopy
Topology Seminar: Spectral Sequences and The Path Fibration
Several Complex Variables Seminar: The Hodge Theorem Part 2
Topology Seminar: The Thom Isomorphism
Topology Seminar: Hochschild Cohomology
Food For Thought: Symmetric Bilinear Forms & 4-manifolds