Narrow Results

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.