Random knots

We introduce a new model for random knots and links, based on the petal projection developed by C. Adams et al. We study the distribution of various invariants of knots and links in this model. We view a knot invariant as a random variable on the set of all petal diagrams with n petals, and ask for its limiting distribution as n --> infinity. We obtain a formula for the limiting distribution of the linking number of a random two-component link. We obtain formulas for all moments of the two most basic Vassiliev invariants of knots, which are related to the Conway polynomial and the Jones polynomial. These are the first precise formulas given for the distributions or moments of invariants in any model for random knots and links.

Joint work with Chaim Even-Zohar, Joel Hass, and Nati Linial.