# Dresden 2003 – wissenschaftliches Programm

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# DY: Dynamik und Statistische Physik

## DY 36: Nonlinear dynamics II

### DY 36.2: Vortrag

### Mittwoch, 26. März 2003, 16:45–17:00, G\"OR/226

**Equilibrium shapes of knots** — •Andreas Hanke^{1,2}, Ralf Metzler^{3,2}, Paul G. Dommersnes^{4}, Yacov Kantor^{5}, and Mehran Kardar^{2} — ^{1}Institute of Theoretical Physics, University of Stuttgart - Pfaffenwaldring 57, D-70550 Stuttgart, Germany — ^{2}Department of Physics, MIT - 77 Mass. Ave., Cambridge, Massachusetts 02139, USA — ^{3}NORDITA - Blegdamsvej 17, DK-2100 Copenhagen, Denmark — ^{4}Institut Curie - 11, rue Pierre et Marie Curie, F-75231 Paris Cedex 5, France — ^{5}School of Physics, Tel Aviv University - IL-69978 Tel Aviv, Israel

In the biological domain, the topology of knotted DNA can be actively modified through energy-consuming enzymes (topoisomerases), the understanding of their detection methods requires knowledge on the equilibrium shapes of knotted polymers.

Given this motivation, we study the equilibrium shapes of prime and composite knots firstly confined to two dimensions, using rigorous scaling arguments originally developed for general polymer networks. For dilute polymers in a good solvent, i.e., above the Theta point, we show that, due to self-avoiding effects, the topological details of prime knots are localised on a small portion of the larger ring polymer. Within this region, the original knot configuration can assume a hierarchy of contracted shapes, the dominating one given by just one small loop. This hierarchy is investigated in detail for the flat trefoil knot.

For polymers in the dense state and at the Theta point, a similar analysis shows, in contrast, that all prime knots are delocalised over the entire ring polymer. A similar behavior is conjectured also for knots in three dimensions. Our results are corroborated by Monte Carlo simulations.