Links with small lattice stick numbers

Abstract

Knots and links have been considered to be useful models for structural analysis of molecular chains such as DNA and proteins. One quantity that we are interested on molecular links is the minimum number of monomers necessary to realize them. In this paper we consider every link in the cubic lattice. Lattice stick number $s_L(L)$ of a link $L$ is defined to be the minimal number of sticks required to construct a polygonal representation of the link in the cubic lattice. Huh and Oh found all knots whose lattice stick numbers are at most 14. They proved that only the trefoil knot $3_1$ and the figure-8 knot $4_1$ have lattice stick numbers 12 and 14, respectively. In this paper we find all links with more than one component whose lattice stick numbers are at most 14. Indeed we prove combinatorically that $s_L(2^2_1)=8$, $s_L(2^2_1 \sharp 2^2_1)=s_L(6^3_2)=s_L(6^3_3)=12$, $s_L(4^2_1)=13$, $s_L(5^2_1)=14$ and any other non-split links have stick numbers at least 15.