Upper Bounds on Linking Numbers of Thick Links

The maximum of the linking number between two lattice polygons of lengths n1, n2 (with n1 ≤ n2) is proven to be of the order of n1(n2) 1 3 . 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 n4/3/π. In the second half of the paper examples are given to show that linking numbers of order n1(n2) 1 3 can be obtained when n1 ≤ n2. When n1 < n2, it is further shown that the maximum of the linking number between these two polygons is bounded by cn1 for some constant c > 0. Finally the maximal total linking number of lattice links with more than 2 components is generalized to k components.