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

TL;DR

This paper extends BFACF-style algorithms to polygons in BCC and FCC lattices, proving ergodicity with respect to knot types and estimating minimal lengths of knotted polygons in these lattices.

Contribution

It generalizes BFACF algorithms to BCC and FCC lattices and establishes their ergodicity with respect to knot types, expanding previous results for cubic lattices.

Findings

Ergodicity classes match knot types in BCC and FCC lattices.

Algorithms estimate minimal lengths of knotted polygons.

Implementation results support theoretical findings.

Abstract

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.