VIGRE Working Group on Khovanov Homology
and the Jones Polynomial of Knots
knot1knot 2Spring 2005

Qi Chen
Sergei Chmutov
Thomas Kerler



Prerequisites
 We will assume a reasonable background in linear algebra and basic abstract algebra (670-672 or equivalent/higher) as well as point set topology and the classification of surfaces from the elementary topology sequence (655, 656).

 Although some familiarity would be useful, we will not assume any knowledge in homology theory. The relevant notions and tools from homological algebra will be introduced and developed from scratch in the context of Khovanov's construction (which is only in form similar to the homology theory used in algebraic topology).  

 Find in the references a few books if you wish to read ahead on homology.


 
References
Talk Slides

Quantum Topology (Invitation to Research, 1/24/05 and 1/31/05)


Books on Modern Knot Theory


Introductory Papers on Khovanov Homology

  
Books on  Homological Algebra

Related and Useful  
    



Outline & Projects
 We will begin with basic introductions to the Jones Polynomial following Chapters 3-5 in Lickorish's book and to Khovanov's homology theory following Bar-Natan's paper. The prerequisites for these are fairly elementary and we plan to have students present them in seminar talks.  Refer to the linked talk below for a general idea.

   This is followed by a more in-depth study of Khovanov's original paper and the surrounding literature.  This will also include acquiring background knowledge in homological algebra as we go along.

   The main project we want to focus on is to study the behavior of natural generalization of Khovanov's chain complexes under the Reidemeister Moves, in order to see whether we an extract more general and geometrical invariants from this.

    Other possible projects are adaptations of Khovanov's homology to knotted graphs, virtual knots and links, Legendrian knots, and their projections on surfaces rather than the plane. There are also various programs available that compute Khovanov homology and which we can adapt and use to explore  our various generalizations.


 
Background & History
  vfr Vaughan Jones' discovery of a new polynomial for knots in 1984 reshaped the landscape of knot theory and, more generally, 3-dimensional topology. Not enough that his discovery was motivated by the theory of van Neuman operator algebras and II-1 subfactors (rather than knot theory itself), in the following  years the Jones Polynomial was rediscovered in areas as diverse as Hopf algebra theory, quantum field theory, quantum groups, integrable lattice models, soliton scattering, category theory, non-commutative geometry,  combinatorial approaches to knots theory etc..

   The profound impact of Jones' contribution earned him the Fields Medal in 1990.   Based on natural generalizations of the underlying algebra, many other such knot invariants have been found since, such as the HOMFLYPT and the Kauffman Polynomial and their "colorings".
lkIt is also the basic ingredient for the celebrated Witten-Reshetikhin-Turaev Invariants of 3-manifolds discovered in 1989.

  One of the great advantages of the Jones Polynomial and its relatives is that they can be computed quite easily and algorithmically from a concrete projection of a knot. Moreover, these new invariants of knots have been proved to encompass and vastly dominate nearly all of the previously known knot invariants. It is generally believed that the Jones Polynomial is the first invariant to be able to detect unknotedness.

 tait Due to its combinatorial nature, however, very little is understood about the geometric content of the Jones polynomial and hence its implications for the  topology of knots. There are some successes, though, such as detecting the non-amphicierality of certain knots and the proof of the Tait Conjecture (see the talk presentation). Nevertheless, from the fact that a powerful invariant as this admits such a natural and universal definition one would expect a much wider range with appropriate geometric and topological interpretations.  

  
In 1999 Mikhail Khovanov, in an attempt to find invariants for 4-manifolds,  discovered a (Z-graded) homology theory associated to knots and links derived from the resolution complexes of their projections. The Jones Polynomial turns out to be themk (graded) Euler characteristic of this homology. The latter is thus both a true refinement of the Jones Polynomial as well as  offers a more geometrical approach to the same. (In fact, several relations to the Heegaard Floer Homology to associated symplectic manifolds have since been established).

 Still, the construction of Khovanov Homology uses mainly algebraic and combinatorial methods which makes it a promising target for attempts to find useful generalizations. Some of these have already been found and successfully applied to topology.