BFACF-style algorithms for polygons in the body-centered and face-centered cubic lattices

In this paper the elementary moves of the BFACF-algorithm for lattice polygons are generalised to elementary moves of BFACF-style algorithms for lattice polygons in the body-centred (BCC) and face-centred (FCC) cubic lattices. We prove that the ergodicity classes of these new elementary moves coincide with the knot types of unrooted polygons in the BCC and FCC lattices and so expand a similar result for the cubic lattice. Implementations of these algorithms for knotted polygons using the GAS algorithm produce estimates of the minimal length of knotted polygons in the BCC and FCC lattices

Polygonal valuations

AbstractWe develop a valuation theory for generalized polygons similar to the existing theory for dense near polygons. This valuation theory has applications for the study and classification of generalized polygons that have full subpolygons as subgeometries

Osculating and neighbour-avoiding polygons on the square lattice

We study two simple modifications of self-avoiding polygons. Osculating polygons are a super-set in which we allow the perimeter of the polygon to touch at a vertex. Neighbour-avoiding polygons are only allowed to have nearest neighbour vertices provided these are joined by the associated edge and thus form a sub-set of self-avoiding polygons. We use the finite lattice method to count the number of osculating polygons and neighbour-avoiding polygons on the square lattice. We also calculate their radius of gyration and the first area-weighted moment. Analysis of the series confirms exact predictions for the critical exponents and the universality of various amplitude combinations. For both cases we have found exact solutions for the number of convex and almost-convex polygons.Comment: 14 pages, 5 figure

An Optimal Algorithm for the Separating Common Tangents of two Polygons

We describe an algorithm for computing the separating common tangents of two simple polygons using linear time and only constant workspace. A tangent of a polygon is a line touching the polygon such that all of the polygon lies to the same side of the line. A separating common tangent of two polygons is a tangent of both polygons where the polygons are lying on different sides of the tangent. Each polygon is given as a read-only array of its corners. If a separating common tangent does not exist, the algorithm reports that. Otherwise, two corners defining a separating common tangent are returned. The algorithm is simple and implies an optimal algorithm for deciding if the convex hulls of two polygons are disjoint or not. This was not known to be possible in linear time and constant workspace prior to this paper. An outer common tangent is a tangent of both polygons where the polygons are on the same side of the tangent. In the case where the convex hulls of the polygons are disjoint, we give an algorithm for computing the outer common tangents in linear time using constant workspace.Comment: 12 pages, 6 figures. A preliminary version of this paper appeared at SoCG 201

Asymptotic Behavior of Inflated Lattice Polygons

We study the inflated phase of two dimensional lattice polygons with fixed perimeter $N$ and variable area, associating a weight $\exp[pA - Jb ]$ to a polygon with area $A$ and $b$ bends. For convex and column-convex polygons, we show that $/A_{max} = 1 - K(J)/\tilde{p}^2 + \mathcal{O}(\rho^{-\tilde{p}})$, where $\tilde{p}=pN \gg 1$, and $\rho<1$. The constant $K(J)$ is found to be the same for both types of polygons. We argue that self-avoiding polygons should exhibit the same asymptotic behavior. For self-avoiding polygons, our predictions are in good agreement with exact enumeration data for J=0 and Monte Carlo simulations for $J \neq 0$. We also study polygons where self-intersections are allowed, verifying numerically that the asymptotic behavior described above continues to hold.Comment: 7 page