The Writhe of Knots in the Cubic Lattice

The writhe of a knot in the simple cubic lattice can be computed as the average linking number of the knot with its pushoffs into four non-antipodal octants. We use a Monte Carlo algorithm to generate a sample of lattice knots of a specified knot type, and estimate the distribution of the writhe as a function of the length of the lattice knots. If the expected value of the writhe is not zero, then the knot is chiral. We prove that the writhe is additive under concatenation of lattice knots and observe that the mean writhe appears to be additive under the connected sum operation. In addition we observe that the mean writhe is a linear function of the crossing number in certain knot families.