Computer Science/Discrete Mathematics Seminar II

A knot is a closed curve embedded in the three-dimensional space, like a rope whose two ends are joined together. As usual in Topology, two knots are equivalent if one can be deformed into the other by continuous moves, where stretching and squeezing are allowed but with no cutting and pasting. Random curves in space and how they are knotted give an insight into the behavior of "typical" knots and links. 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 work with Hass, Linial and Nowik, we study random knots based on petal projections, developed by Adams et al. (2012). We have found explicit formulas for the distribution of the linking number of a two component link. We also find formulas for the moments of two finite type invariants of knots, the Casson invariant and another coefficient of the Jones polynomial. The talk will begin with a short introduction to Knot Theory, including all necessary background, and explanation of the above terms. If time permits, some other models of random knots will be discussed.