The BFACF algorithm and knotted polygons

Phase diagrams of knotted and unknotted ring polymers

“…Such a proof would not be trivial, but, applied to ring polymers, pull moves closely resemble moves of the BFACF algorithm, a non-length-conserving method used for lattice ring polymers. BFACF has been shown to have ergodicity classes which are the knot classes [37]. In the absence of a counter-example, we make the assumption that the pull move set is ergodic, within each knot class.…”

Low Loss Intersection of Si Photonic Wire Waveguides

“…This is consistent with α(τ ) = α(∅) + 1 if τ is a prime knot. Whittington 1990), so that knotted conformations have little or no effect on the estimated value of µ when this is computed over all polygon conformations with less than 10 3 or 10 4 edges. (In series enumerations of polygons on Z 3 by Guttmann (1989) there are precisely no knots encountered.)…”

“…The conjecture has been settled in the affirmative for a lattice model (Sumners andWhittington 1988, Pippenger 1989) and for some continuum models (Diao et al 1994). There has also been a considerable amount of numerical work on this problem, attempting to estimate the rate of increase of the knot probability with length (Vologodskii et al 1974, Michels and Wiegel 1984, 1986, Janse van Rensburg and Whittington 1990, Koniaris and Muthukumar 1991, Deguchi and Tsurusaki 1994. The problem has also been investigated experimentally for circular DNA molecules (Shaw andWang 1993, Rybenkov et al 1993).…”

The compressibility of minimal lattice knots

“…The GAS algorithm can be implemented in the SC lattice on polygons of given knot type K using the BFACF elementary moves [1,2] to implement the atmospheric moves [15,16]. This implementation is irreducible on classes of polygons of fixed knot type [17].…”