Random graph embeddings

May 23, 2023
Knots occur in all kinds of systems, including graphs. We're used to finding knots in loops, but what about knotted theta curves? Or knotted graphs? In this talk, we discuss a general framework for constructing random embeddings of arbitrary graphs. This is a conditional probability problem, but setting up the conditions correctly requires some ideas from algebraic topology. We'll view the problem as embedding a simplicial 1-complex and see that the embedding data is (in a precise sense) dual to the simplicial homology of the 1-complex. This will enable us to make some new exact calculations for random knots.
Supplementary Materials