The complexity of lattice knots

A family of polygonal knots Kn on the cubical lattice is constructed with the property that the quotient of length L(Kn) over the crossing number Cr(Kn) approaches zero as L approaches infinity. More precisely Cr(Kn) = O(L(Kn)43). 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 ⩽ 3.2L43.