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Algebraic Knot Theory
University of Copenhagen, October 9, 2008
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.
"God created the knots, all else in topology is the work of mortals."
I don't understand...
For some of these points "I don't understand" just means "Dror doesn't
understand". For some others add "and probably nobody else does, either".
They are not, of course, of equal difficulty and/or importance.
- Who needs "trivalent" in "knotted trivalent graphs"?
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.
- The notion of "framing".
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.
- Configuration space integrals / perturbative Chern-Simons theory for
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.
- The Alexander polynomial.
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.
- The Lieberum "gl(1|1)" associator.
- Must associators be so hard? Why?
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.
- The relationship between "genus" and "finite type".
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:
- Gg 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.
- Is there anything else that annihilates Gg? Or
better, are there any other finite type invariants that detect genus, beyod
the coefficients of the Conway polynomial?
- The relationship between "gropes" and "finite type".
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"
- TG-ideals / internal quotients.
- What exactly are TG-algebras? What are the syzygies among the
relations between their generators?
- Virtual knots.
- Quantum groups(!)
- 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"?