Lattice stick number 15 is unattainable for non-splittable links

Abstract

In this paper, we explore mathematical links, defined as closed curves embedded in 3D space. Knot theory studies these structures, which also occur in real-world biopolymers like DNA. Lattice links are links in the cubic lattice. For scientific simulations or statistical studies, links are simplified to lattice links. The lattice stick number, denoted as s L (K), is the minimum number of lattice sticks needed to represent a link K in the cubic lattice. In previous study, it was shown that only two non-trivial knots and six non-splittable links have s L ≤ 14: specifically, s L ( 2 1 2 ) = 8 , s L ( 3 1 ) = s L ( 2 1 2 ♯ 2 1 2 ) = s L ( 6 2 3 ) = s L ( 6 3 3 ) = 12 , s L ( 4 1 2 ) = 13 , and s L ( 4 1 ) = s L ( 5 1 2 ) = 14 . Recent study has further revealed that no knot can have s L = 15. In this paper, we prove that lattice stick number 15 is not attainable for non-splittable links. As a corollary, eleven non-splittable links with s L =16 are presented.