UPPER BOUNDS ON LINKING NUMBERS OF THICK LINKS

Abstract

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 n4/3/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 [Formula: see text]. When [Formula: see text], it is further shown that the maximum of the linking number between these two polygons is bounded by [Formula: see text] for some constant c > 0. Finally the maximal total linking number of lattice links with more than 2 components is generalized to k components.