Counting knots and links

Andrew Rechnizter, University of British Columbia:

Counting knots and links

Abstract: Recently a great deal of attention from biologists has been directed to understanding the role of knots in perhaps the most famous of long polymers - DNA. In order for our cells to replicate, they must somehow untangle the approximately two metres of DNA that is packed into each nucleus. Biologists have shown that DNA of various organisms is non-trivially knotted with certain topologies preferred over others. The aim of our work is to determine the "natural" distribution of different knot-types in random closed curves and compare that to the distributions observed in DNA.

Our tool to understand this distribution is a canonical model of long chain polymers - self-avoiding polygons (SAPs). These are embeddings of simple closed curves into a regular lattice. The exact computation of the number of polygons of length n and  fixed knot type K is extremely difficult - indeed the current best algorithms can barely touch the first knotted polygons. Instead of exact methods, in this talk I will describe an approximate enumeration method - is a generalisation of the famous Rosenbluth method for simulating linear polymers. Using this algorithm we have  uncovered strong evidence that the limiting distribution of different knot-types is universal and is quite different from that found in DNA. I will also discuss some recent extensions of this work from knots to links.