Lattice stick number 15 is unattainable for non-splittable links

This is the authors' abstract. We don't add key points for this paper.

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.