Counting knots - 28 trefoils and 1 figure-eight
|What||Arbeitsgemeinschaft Diskrete Mathematik|
from 15:15 to 16:45
|Where||TU Institut für Diskrete Mathematik und Geometrie, Freihaus, grüner Turm (A), 8. Stock, Dissertantenr., Wiedner Hauptst. 8-10, 1040 Wien|
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Andrew Rechnitzer (University of British Columbia, Vancouver)
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. Unfortunately, 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 of exactmethods, in this talk I will describe an approximate enumeration method - which we call the GAS algorithm. This 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. Our data shows that a long closed curve is about 28 times more likely to be a trefoil than a figure-eight, and that the natural distribution of knots is quite different from those found in DNA.