The curvature of lattice knots

A result of Milnor [1] states that the infimum of the total curvature of a tame knot K is given by 2 pi mu(K), where mu(K) is the crookedness of the k not K. It is also known that mu(K)=b(K), where b(K) is the bridge index of K [2]. The situation appears to be quite different for knots realised as po lygons in the cubic lattice. We study the total curvature of lattice knots by developing algebraic techniques to estimate minimal curvature in the cub ic lattice. We perform simulations to estimate the minimal curvature of lat tice knots, and conclude that the situation is very different than for tame knots in R-3.