 CARMA COLLOQUIUM
 Speaker: Dr Andrew Rechnitzer, UBC
 Title: Counting knots
 Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 4:00 pm, Thu, 31^{st} Oct 2013
 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 nontrivially
knotted with certain topologies preferred over others. The
aim of our work is to determine the "natural" distribution of
different knottypes 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  selfavoiding 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  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 knottypes is universal. Our data shows
that a long closed curve is about 28 times more likely to be a
trefoil than a figureeight, and that the natural distribution
of knots is quite different from those found in DNA.
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