Narrow Results

For any given knot K, a thick realization K₀ of K is a knot of unit thickness which is of the same knot type as K. In this paper, we show that there exists a family of prime knots {K_n} with the property that Cr(K_n)→∞(as n→∞) such that the arc-length of any thick realization of K_n will grow at least linearly with respect to Cr(K_n).

The vertex distortion of a lattice knot is the supremum of the ratio of the distance between a pair of vertices along the knot and their distance in the ${\ell }_1$-norm. Inspired by Gromov, Pardon and Blair–Campisi–Taylor–Tomova, we show that results about the distortion of smooth knots hold for vertex distortion: the vertex distortion of a lattice knot is 1 only if it is the unknot, and there are minimal lattice-stick number knot conformations with arbitrarily high distortion.

The first two authors introduced vertex distortion of lattice knots and showed that the vertex distortion of the unknot is 1. It was conjectured that the vertex distortion of a knot class is 1 if and only if it is trivial. We use Denne and Sullivan’s lower bound on Gromov distortion to bound the vertex distortion of non-trivial lattice knots. This bounding allows us to conclude that a knot class has vertex distortion 1 if and only if it is trivial. We also show that vertex distortion does not have a universal upper bound and provide a vertex distortion calculator.

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.