The complexity of lattice knots

A family of polygonal knots K-n on the cubical lattice is constructed with the property that the quotient of length L(K-n) over the crossing number Cr(K-n) approaches zero as L approaches infinity. More precisely Cr(K-n) = O(L(K-n)(4/3)). It is shown that this construction is optimal in the sense that for any knot K on the cubical lattice with length L and Cr crossings Cr less than or equal to 3.2 L-4/3. (C) 1998 Elsevier Science B.V. All rights reserved.