- 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, 31st 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 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 - 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.
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