The writhe of polygons on the face-centred cubic lattice

For polygons on the simple cubic lattice there is an important theorem due to Lacher and Sumners which shows that the writhe of a polygon is the avera ge of the linking numbers of the polygon and its pushoffs in four particula r directions. This implies that the writhe of a polygon on the simple cubic lattice is always rational. We prove a related theorem for the face-centre d cubic (fcc) lattice but show that, on this lattice, the probability that a polygon of length n has irrational writhe tends to unity as n tends to in finity. In addition, we show that the expectation of the absolute value of the writhe increases at least as fast as root n for large n.