Title: Invariants of Random Knots.
Abstract:
Random curves in space and how they are knotted give an insight into the behavior of "typical" knots and links, and are expected to introduce the probabilistic method into the mathematical study of knots. They have been studied by biologists and physicists in the context of the structure of random polymers. There have been many results obtained via computational experiment, but few explicit computations.
In the talk, I will focus on a new, combinatorial model for generating curves at random, based on petal projections, developed by Adams et al. (2012). In work with Hass, Linial and Nowik, we found explicit formulas for the distribution of the linking number of a random two-component link. We also found formulas for the moments of two finite type invariants of knots, the Casson invariant and another coefficient of the Jones polynomial. These are the first precise formulas of this sort in any model for random knots or links.
If time permits, some other models of random knots will be discussed.
All necessary background, and the above terms will be explained.
Joint work with Joel Hass, Nati Linial, and Tahl Nowik.