Upper bounds on the minimum length of cubic lattice knots

Abstract

Knots have been considered to be useful models for simulating molecular chains such as DNA and proteins. One quantity that we are interested in is the molecular knot and the minimum number of monomers necessary to realize a knot. In this paper, we consider every knot in the cubic lattice. In particular, the minimum length of a knot indicates the minimum length necessary to construct the knot in the cubic lattice. Diao introduced this term (he used 'minimum edge number' instead) and proved that the minimum length of the trefoil knot 3(1) is 24. Also theminimum lengths of the knots 4(1) and 5(1) are known to be 30 and 34, respectively. In this paper we find a general upper bound of theminimum length of a nontrivial knot K, except the trefoil knot, in terms of the minimum crossing number c(K). The upper bound is 3/2c(K)(2) + 2c(K) + 1/2. Moreover, if K is a nonalternating prime knot, then the upper bound is 3/2c(K)(2) -4c(K) + 5/2. Ourwork can be considered a direct consequence of the results obtained by the authors in Hong et al (2012 arXiv: 1209.0048). Furthermore, if K is (n + 1, n)-torus knot, then the upper bound is 6c(K) + 2 root c(K) + 1 + 6.