Abstract
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 ⩽ 3.2L 4 3 .
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