© | Dror Bar-Natan: Academic Profile:

Summary of Proposal, 2007

(Submitted to the NSERC)

In the space provided below, state the objectives of the proposed research program and summarize the scientific approach, highlighting the novelty and expected significance of the work to a field or fields in the natural sciences and engineering. [...] Your summary must not exceed forty-five lines on the printed copy.

Over the next five years I plan to pursue following three projects.

   * Algebraic Knot Theory. For many years now the Kontsevich integral Z (a universal finite type invariant of knots and links) is appreciated for its strength. It is stronger than all known "quantum invariants" taken together. But only recently I understood that that might not be where the real power of this invariant lies: Z beautifully extends to an invariant of knotted trivalent graphs which is well behaved under certain natural operations defined on graphs - edge deletion, "unzipping" and connected sums. Several well known and hard-to-detect properties of knots are "definable" using these operations, including the knot genus, unknotting numbers and the property of being a ribbon knot. Thus, at least in principle, each of these properties can be translated to a simple algebraic "equation" involving Z of the knot being studied. But turning this principle into results is a five-year project. We still have to identify and study appropriate quotients of the target space of Z which make Z computable to all orders. Every classical knot polynomial (Alexander's, Jones', etc.) defines such a quotient, and there are many other quotients to choose from, but even the simplest of these quotients, corresponding to a canonical extension of the Alexander polynomial to graphs, is poorly understood. See http://www.math.toronto.edu/~drorbn/Talks/Aarhus-0706.
   * Knot Theoretic Algebra. My paper "On Associators and the Grothendieck-Teichmuller Group" indicates strongly that the right context for understanding Drinfel'd's theory of formal associators is a certain category of braid groups and operations mapping such braids groups to each other. There is strong evidence that the theory of quantum groups (specifically, quantum universal enveloping algebra and/or quasi-triangular Hopf algebras) should be related in a similar manner to virtual knots and braids and operations among them. Indeed, one of the starting points of the theory of quantum groups is the quantization of Lie bialgebras, and the "universal" diagrammatic theory underlying Lie bialgebras is the same as the diagrammatic theory that underlies finite type invariants of virtual knots/braids. (Compare with the well-understood relationship between knots, chord diagrams and Lie algebras). Over the grant period I plan to fully understand a universal theory of quantum groups as a natural object within the context of virtual knot theory. See http://www.math.toronto.edu/~drorbn/Talks/Kyoto-0705/ and ---/Tianjin-0707/.
   * Computations in Knot Theory and the Knot Atlas. I Plan to continue contributing to the computational package KnotTheory` and to the Knot Atlas. Both projects were founded by me but by now have received contributions by many others (especially S. Morrison). See http://katlas.org.

What about Khovanov homology? I made significant contributions to the highly fashionable subject of Khovanov homology (in fact, while Khovanov is definitely the father of the field, I share the credit for making it fashionable...). Yet at the moment I don't feel mature enough to study this topic any further. I'd rather "categorify" knot invariants only after I properly understand the "algebra" on which they ought to be defined (in the sense of my first project above). And how can I even start categorifying other aspects of the theory of quantum groups, when in my opinion this theory in itself is so poorly understood (at least in the sense of my second project)? With luck, at the end of this grant period I will be ready to return to Khovanov homology and categorification in general.

In science, though, the predicted is always less interesting than the unpredictable. With luck, at least some of my work in the next five years will be on topics I haven't yet heard of.