1999-09-17
11:15 at CAB D28

Knot complexity measures applied to polymers simulated as polygonal curves

Chantal Oberson

EPFL, Lausanne

 In this talk, I will summarize the work performed for my mathematics Ph.D. dissertation. The goal is to use the mathematical theory of knots to study the entanglement complexity of polymers, simulated as polygonal curves. Knot theory is an area of algebraic topology, which mainly deals with classification of the different knots and links and therefore provides one splendid tool : the knot invariants. I have chosen one of them, the HOMFLY polynomial, to classify the knots obtained by simulation of polymers. Other invariants, like genus and determinant, have been used as good measures of knot complexity. These functions assign a numerically-valued complexity to knots, once they have been identified by the HOMFLY polynomial. More precisely, I have written a program, which, for a given polymer and a corresponding rotational isomeric state model, generates random polygonal curves with 10,000 backbone bonds, identifies the knots obtained, gives them a complexity, which is then averaged over the whole sample of 5,000 chains generated. One crucial result is that all the selected complexity measures, although very different in nature, respect the same classification of the polymers as the proportion of trivial knots in the sample. I have then studied the approximation of a polymer chain as a Kuhn chain, to see if it was satisfactory from an entanglement point of view. The program could also be used to investigate meaningful relationships between certain physical properties of polymers and their entanglement degree.