**abstract:**
Self-avoiding polygons are simple closed curves embedded in a regular lattice.
They belong to a family of objects, known collectively as lattice animals, that lie at
the heart of lattice models of polymers, magnets and other phenomena. This large
family of discrete structures has been a source of combinatorial problems for over
50 years and many basic questions remain stubbornly unsolved. For example, the
simple enumerative question - "How many self-avoiding polygons are there of
length n?" is currently best answered by an algorithm that requires exponential time
and memory.
The question of counting polygons on the cubic lattice is enriched by topology -
self-avoiding polygons have well defined knot types. 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. In
this talk I will discuss a different approach to the problem. Instead of of exact
methods, we have used an approximate enumeration method - which we call the
GAS algorithm. This is a generalisation of the famous Rosenbluth method for
sampling self-avoiding walks.
Using this algorithm we have estimated the number of polygons of different lengths
and knot types on three different cubic lattices. These give direct evidence that the
asymptotic growth of the number of polygons of a fixed knot type K is simply
related to the growth of the number of unknotted polygons and the number of prime
components in K. We have also studied the relative frequencies of different knots -
for example, the ratio of the number of trefoils to the number of figure-eights. We
see strong evidence that these ratios are universal suggesting that a very long closed
curve is about 27 times more likely to be a trefoil than a figure-eight.

Thu 9 Jun, 11:30 - 12:30, Aula Dini

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