Thoughts on lattice knot statistics

• DOI: 10.1007/s10910-008-9364-9
• Publication Date: 2008-08-26T12:51:12Z
• Authors: E. J. Janse van R...
• Keywords: Lattice (music)

Lattice knot statistics, or the study of knotted polygons in the cubic lattice, gained momentum in 1988 when the Frisch-Wasserman-Delbruck conjecture was proven by Sumners and Whittington (J Phys A Math Gen 21:L857–861, 1988), and independently in 1989 by Pippenger (Disc Appl Math 25:273–278, 1989). In this paper, aspects of lattice knot statistics are reviewed. The basic ideas underlying the study of knotted lattice polygons are presented, and the many open problem are posed explicitly. In addition, the properties of knotted polygons in a confining slab geometry are explained, as well as the Monte Carlo simulation of knotted polygons in \({{\mathbb Z}^3}\) and in a slab geometry. Finally, the mean behaviour of lattice knots in a slab are discussed as a function of the knot type.