Abstract
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.
Funder
National Research Foundation of Korea