Knots with small lattice stick numbers
Youngsik Huh, Seungsang Oh

TL;DR
This paper proves that only the trefoil and figure-8 knots have lattice stick numbers less than 15, confirming previous numerical estimates through rigorous mathematical proof.
Contribution
It provides a rigorous mathematical proof identifying the only knot types with lattice stick numbers under 15, specifically the trefoil and figure-8 knots.
Findings
Trefoil knot and figure-8 knot are the only knots with lattice stick number less than 15.
Validated previous numerical estimations with a formal proof.
Established a boundary for lattice stick numbers of knots.
Abstract
The lattice stick number of a knot type is defined to be the minimal number of straight line segments required to construct a polygon presentation of the knot type in the cubic lattice. In this paper, we mathematically prove that the trefoil knot and the figure-8 knot are the only knot types of lattice stick number less than 15, which verifies the result from previous numerical estimations on this quantity.
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