Minimal knotted polygons in cubic lattices

作者：van Ren**urg E J Janse^*; Rechnitzer A

来源：Journal of Statistical Mechanics: Theory and Experiment , 2011, P09008.

DOI：10.1088/1742-5468/2011/09/P09008

• topology and combinatorics
• classical Monte Carlo simulations
• random graphs
• networks
• polymers
• copolymers
• polyelectrolytes and biomolecular solutions

In this paper we examine numerically the properties of minimal length knotted lattice polygons in the simple cubic, face-centered cubic, and body-centered cubic lattices by sieving minimal length polygons from a data stream of a Monte Carlo algorithm, implemented as described in Aragao de Carvalho and Caracciolo (1983 Phys. Rev. B 27 1635), Aragao de Carvalho et al (1983 Nucl. Phys. B 215 209) and Berg and Foester (1981 Phys. Lett. B 106 323). The entropy, mean writhe, and mean curvature of minimal length polygons are computed (in some cases exactly). While the minimal length and mean curvature are found to be lattice dependent, the mean writhe is found to be only weakly dependent on the lattice type. Comparison of our results to numerical results for the writhe obtained elsewhere (see Janse van Rensburg et al 1999 Contributed to Ideal Knots (Series on Knots and Everything vol 19) ed Stasiak, Katritch and Kauffman (Singapore: World Scientific), Portillo et al 2011 J. Phys. A: Math. Theor. 44 275004) shows that the mean writhe is also insensitive to the length of a knotted polygon. Thus, while these results for the mean writhe and mean absolute writhe at minimal length are not universal, our results demonstrate that these values are quite close the those of long polygons regardless of the underlying lattice and length.

• 出版日期2011-9