Algebraic Knot Theory

Abstract. The right objects of study in algebraic topology are not spaces, but rather, "spaces and maps between them". In a similar spirit I will argue that the right things to study in knot theory are not knots, but rather, "knotted trivalent graphs", as in the world of knotted trivalent graphs (and the basic operations between them) many interesting properties of honest knots become "definable". Thus I find myself again studying the good old Kontsevich integral - the best example I know of an algebraic knot theory - but my perspective this time is completely different. (more)

Some propaganda...

"God created the knots, all else in topology is the work of mortals." Leopold Kronecker (modified)

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I don't understand... (more)

• Who needs "trivalent" in "knotted trivalent graphs"? (more)
  Topologically speaking knotted graphs are just tangles - take your graph, choose a maximal tree in it, contract it to a small disk, and what you have is a tangle on the outside of that disk. So a knotted trivalent graph is just a tangle with a bit of further combinatorial information. But for reasons I don't understand, this further combinatorial information is necessary in order to define perturbative invariants. (less)
• The notion of "framing". (more)
  There are (at least) two (equivalent) definitions for framing a knot: one has to do with finding a section of the normal bundle of the knot (or just turning the knot into a band knot); the other has to do with extending the Gauss map from the circle to the disk (even more). It is the latter definition which is more relevant for configuration space integrals, and I don't know how to extend it to knotted graphs. (less)
• Configuration space integrals / perturbative Chern-Simons theory for knotted graphs. (more)
  The definitions are obvious. Invariance and good behaviour under the basic operations of knotted trivalent graphs are plausible, but yet unproven. And I don't understand how the up-to-twists indeterminacy of the tetrahedron (i.e., the Drinfel'd associator) comes about. (less)
• The Alexander polynomial. (more)
  I know that there is an extension of the Alexander polynomial to knotted trivalent graphs which is well-behaved under the basic operations on those. But I don't know what it is. (less)
• The Lieberum "gl(1|1)" associator. (more)
  Here it is: (taken from arXiv:math.QA/0204346)

  

  Do you understand it? (less)

• Must associators be so hard? Why? (more)
  Associators exist, but there does not seem to be a closed form formula for one. Is this somehow because they are related to somehow to hard problems in Galois theory, as is suggested by the papers by Drinfel'd? Or should we accept that functional equations tend to have no closed form solutions? If the latter, we should then be surprised by the fact that the equations defining (say) R-matrices in quantum groups (which are of the same general nature) are in fact soluble. (less)
• The relationship between "genus" and "finite type". (more)
  It is rather easy to interpret the map at the level of chord diagrams, and thus write a description of a subspace Gg of the space of all chord diagrams that contains the Kontsevich integrals of all knots of genus at most g. But it is still hard to actually analyze that subspace:

  1. G_{g must be annihilated by high coefficients of the Conway polynomial. I've verified it for some low coefficients but I still don't know how to prove that in general.}
  2. Is there anything else that annihilates G_{g? Or better, are there any other finite type invariants that detect genus, beyod the coefficients of the Conway polynomial? (less)}
• The relationship between "gropes" and "finite type". (more)
  Something similar to the genus story may exist here, but I know nothing yet. Learn more by web-searching for "grope" or "gropes" along with "Conant" or "Teichner". (less)
• TG-ideals / internal quotients.
• What exactly are TG-algebras? What are the syzygies among the relations between their generators?
• Virtual knots.
• Quantum groups(!) (more)
  Also see Talks: Kyoto-0705. (less)
• The work of Etingof and Kazhdan.
• The polynomiality of knot polynomials.
• Functional equations.
• The Eilenberg-Zilber theorem.
• Which other interesting classes of knots/links are "TG-definable"?