# Random Knots are Knotted

## Chaim Even Zohar, University of California Davis

## Date and time: 6pm, Tuesday, February 26, 2019

## Place: Courant Institute, WWH1314

A variety of random knot models have been proposed and investigated by
mathematicians and biologists, who are interested in such models to
study the structure of polymers and to compare their knot types to
those that arise by random processes. More broadly, the probabilistic
approach gives an insight into the behavior of “typical” knots and
links. The talk will begin with a review of this area.

A desired property for such a random model is that the probability of
obtaining every specific knot type decays to zero as the typical
complexity of the knot increases. Past approaches to establish this
property, in several random models, rely on the prevalence of
localized connect summands. However, this phenomenon is not clear in
other models that exhibit, in a sense, “a large step length” or
“spatial confinement.”

In joint work with Joel Hass, Nati Linial, and Tahl Nowik, we use
finite type invariants and a coupling argument to establish this
property for random knots that arise from petal projections. We expect
our methods to extend to other well-studied knot models, in which
local entanglements are similarly believed to be rare, such as random
grid diagrams and uniform random polygons.