The shapes of self-avoiding polygons with torsion

We consider self-avoiding polygons on the simple cubic lattice with a torsion fugacity. We use Monte Carlo methods to generate large samples as a function of the torsion fugacity and the number of edges in the polygon. Using these data we investigate the shapes of the polygons at large torsion fugacity and find evidence that the polygons have substantial helical character. In addition, we show that these polygons have induced writhe for any non-zero torsion fugacity, and that torsion and writhe are positively correlated. There is considerable interest in geometrical measures of entanglement complexity of selfavoiding walks, and related structures such as polygons and ribbons, and these ideas have proved to be especially useful in describing models of double stranded polymers such as DNA (Bauer et al 1980). Two useful measures of geometrical entanglement complexity for a simple closed curve in three dimensions are writhe and torsion. Torsion characterizes the local helicity of the curve while writhe captures information about the non-local crossings of the curve with itself. The writhe of a polygon in Z3 can be conveniently calculated by making use of a theorem due to Lacher and Sumners (1991) which shows that the writhe is the mean of the linking number of the polygon with its pushoffs into four mutually nonantipodal octants. This result is an essential ingredient in the proof that the expected value (over all n-gons) of the absolute value of the writhe of polygons in Z3 increases at least as fast as √ n (Janse van Rensburg et al 1993). If the polygon has fixed knot type then the expected value of the writhe depends only weakly on n but is a function of the knot type of the polygon (Janse van Rensburg et al 1997), and is zero if the knot is achiral. For a smooth curve in R3 one can define the torsion in terms of a line integral (see, for example, Struik 1988) but, since we shall be concerned with piecewise linear curves, we define it in terms of dihedral angles (Alexandrov and Reshetniyak 1989). A polygon is made up of a sequence of line segments. Each consecutive triple of line segments defines a dihedral angle about the central segment of the triple. If the three line segments are coplanar this dihedral angle is either 0 or π . If they are non-coplanar it is ±π/2. A positive dihedral angle is a dihedral angle of π/2 and a negative dihedral angle is a dihedral angle of −π/2. We associate a quantity τi with the ith line segment, and set τi = ±1 according to whether ¶ Permanent address: Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada. 0305-4470/97/200693+06$19.50 c © 1997 IOP Publishing Ltd L693 L694 Letter to the Editor the corresponding dihedral angle is positive or negative, and zero otherwise. We define the torsion of the polygon as