Narrow Results

The main theorem in this article shows that the link cannot be realized with fewer than 28 steps on the cubical lattice. There are at least two very different 28-step realizations of on the cubical lattice in which the length of the components of the link differs. The two 28-step realizations of do not have minimal curvature. A realization with a minimal curvature of 6.5 π of the link with 30 steps is also shown.

The maximum of the linking number between two lattice polygons of lengths n₁, n₂ (with n₁ ≤ n₂) is proven to be the order of n₁ (n₂)^⅓. This result is generalized to smooth links of unit thickness. The result also implies that the writhe of a lattice knot K of length n is at most 26 n^4/3/π. In the second half of the paper examples are given to show that linking numbers of order n₁ (n₂)^⅓ can be obtained when . When , it is further shown that the maximum of the linking number between these two polygons is bounded by for some constant c > 0. Finally the maximal total linking number of lattice links with more than 2 components is generalized to k components.