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 in for molecular links is the minimum number of monomers necessary for realizing them. In this paper we consider every link in the cubic lattice. The lattice stick number sL(L) of a link L is defined to be the minimum 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 31 and the figure-eight knot 41 have lattice stick numbers of 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 , , , and any other non-split links have stick numbers of at least 15.

Circuit presentation and lattice stick number with exactly four z-sticks

“…Furthermore Diao and Ernst [4] found a general lower bound in terms of crossing number c(K) which is s L (K) ≥ 3 c(K) + 1+3 for a nontrivial knot K. Hong, No and Oh [5] found a general upper bound s L (K) ≤ 3c(K)+2, and moreover s L (K) ≤ 3c(K)−4 for a non-alternating prime knot. They [6] also showed that s L (2 2 1 ) = 8, s L (2 2 1 ♯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.…”

“…Furthermore Diao and Ernst [4] found a general lower bound in terms of crossing number c(K) which is s L (K) ≥ 3 c(K) + 1+3 for a nontrivial knot K. Hong, No and Oh [5] found a general upper bound s L (K) ≤ 3c(K)+2, and moreover s L (K) ≤ 3c(K)−4 for a non-alternating prime knot. They [6]…”

Escher squares and lattice links

“…A lattice link is a link L ⊂ G 3 with one or more components. Much of the research on lattice links focuses on finding lattice stick numbers, the minimal number of line segments (possibly containing more than one edge) necessary to construct a link type as a lattice link [1,3,5,6,7]. One of the first results was that of Diao [3], who proved that the lattice stick number of the trefoil is 12.…”

“…Adams et al [1] found lattice stick numbers for various knots and links, including all (p, p + 1)−torus knots. Hong, No and Oh [5] found all links with more than one component whose lattice stick numbers are at most 14. Lattice links have played an important role in simulating various circular molecules [8].…”

Lattice stick number of spatial graphs

“…Adams et al [1] found that s L (T p,p+1 ) = 6p for p ≥ 2 where T p,p+1 is a (p, p + 1)-torus knot. All links with more than one component with lattice stick number at most 14 were found by Hong, No and Oh [6]. Janse Van Rensburg and Promislow [11] proved that s L (K) ≥ 12 for nontrivial knots K. Adams et al [1] found an upper bound s L (K) ≤ 6c(K) − 4.…”