curso por Alexei Sossinsky (Universidade Independente de
Moscovo, Rússia)
Abstract:
The mathematical theory of knots and links, which
studies non-intersecting and non-self-intersecting curves
in space, is at least 150 years old, but has recently
surged to the forefront of science, attracting not only
mathematicians and physicists, but also experts in biology
and chemistry, and in some branches of engineering as well.
In the decade 1985--1995, no less than four mathematicians
who contributed to the theory were awarded the Fields Medal
(the "Nobel Prize for mathematicians"): Vaughan Jones (New
Zealand), Edward Witten (USA), Vladimir Drinfeld (Ukraine),
Maxim Kontsevich (Russia).
Surprizingly, although many of the original papers
involve some very sophisticated mathematics, there now
exists a very elementary exposition of the main results, so
that no advanced mathematical prerequisites will be needed
to follow the course. In it, there will be lots of very
visual geometry (pictures and even computer-generated
animations), some absolutely elementary algebra (adding and
multiplying polynomials, linear spaces, permutation
groups), and a new kind of calculus (which may be called
"games with little diagrams").
The main protagonists, however, will be several kinds of
invariants, i.e., algebraic entities allowing to easily
solve difficult classification problems concerning the main
three-dimensional geometric objects of the course --
knots, links, braids.
Tentative plan:
The geometry and arithmetic of knots:
knot diagrams and equivalence of knots via Reidemeister
moves, decomposition into prime knots, knot tables,
Conway axioms for the Alexander polynomial.
Braids and their relationship with knots and links:
encoding braids by standard generators, Artin theorem
on the braid group, Alexander theorem (any knot is the
closure of a braid), braid comparison algorithms.
The Kauffman bracket and the Jones polynomial:
Kauffman's imaginary statistical model and his bracket,
definition of the Jones polynomial via the Kauffman
bracket, properties of the Jones polynomial, proof of
the Tait conjectures on alternating knots.
Elementary theory of Vassiliev invariants:
Axioms for Vassiliev invariants, the one-term and
four-term relations, the bialgebra of chord diagrams,
Kontsevich theorem (isomorphism of the Vassiliev
(bi)algebra and the (bi)algebra of chord diagrams).
Relationship with physics and biology:
the Potts water-ice model and the Jones polynomial,
the Conway moves and DNA.
Reference: The Knot Book by Colin Adams, AMS (2004).
