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 average of the linking numbers of the polygon and its pushoffs in four particular 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-centred 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 infinity. In addition, we show that the expectation of the absolute value of the writhe increases at least as fast as ( n ) 1/2 for large n .