Abstract:
The talk will address classical knot theory, i.e. the theory of closed 1-dimensional submanifolds of the three-space, the main problem of which is of course their classification.
There exists a formal (algorithmic) solution to this problem, which amounts to a theoretical possibility to assign a natural number to each topological link type so that some algorithm will produce, in finite time, this number for any given link. This solution (which required efforts of many mathematicians) is based on the works of Wolfgang Haken who constructed in 1961 the first algorithm for recognizing the unknot.
His method and algorithms based on it deal not with the link itself but with the manifold obtained by drilling out a tubular neighborhood of the link from the three-sphere.
In the beginning of 2000's it was discovered by the speaker that the unknot can be recognized in a more “natural” way, just by untangling it, if the knot is presented by a so called rectangular diagram. By untangling we mean an application of a sequence of certain elementary transformations that do not increase the complexity of the diagram. It turns out that a rectangular diagram of the unknot can always be untangled completely.
It is tempting to extend this approach to arbitrary knots, just transforming them to a “canonical form” by untangling, but this does not work directly, since non-trivial knots often have more than one rectangular presentation that cannot be simplified further.
Already in 2003 William Menasco drew author's attention to a connection between rectangular diagrams and Legendrian knots, but only recently, in a joint research with Maxim Prasolov, we discovered a way to use this connection efficiently. In particular, the problem of describing non-simplifiable rectangular diagrams was tied to classification of Legendrian knots of a fixed topological type.
We also extended the rectangular presentation formalism to surfaces, which gave us a powerful tool for distinguishing topologically equivalent Legendrian knots. To date this often appears very difficult even for knots with small crossing number (say, just six).
I will survey the above mentioned results in the talk. All the necessary definitions will be given.
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