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Knots and Coxeter Groups

Dylan Burke, Geoffrey Cuff-Chartrand, Malors Espinosa, Mateusz Kazimierczak, Mohammadamin Mobedi

TL;DR

This work investigates knot construction within the affine Coxeter complex of type $\widetilde{B}_3$ by using threefold rotational symmetry. It introduces and analyzes length invariants $l(K)$, $l(\widetilde{B}_3)$, and $l_3(K)$, proving $l(\widetilde{B}_3)\le 40$ and that the smallest threefold-symmetric knot length is $l_3(\widetilde{B}_3)=42$, realized by a symmetric trefoil with $l_3(3_1)=14$. By translating cubic-lattice pieces into $\widetilde{B}_3$ galleries through bar-corner gluings and $TNB$ frame methods, the authors construct explicit order-$3$ galleries for knots $9_{35}, 9_{40}, 9_{41}, 9_{47}$, providing concrete bounds on their threefold-symmetric knot lengths and detailing the translation between lattices. The paper culminates with a sequence of constructive techniques and three open questions about triviality criteria, symmetry-minima interactions, and whether the trefoil is always the minimal knot under symmetric stick constraints. These results deepen understanding of knotting within Coxeter lattices and offer algorithmic tools for generating symmetric knotted galleries.

Abstract

In this paper we study knots created by galleries in the affine Coxeter complex of type \widewedge{B3}. We bound the stick number by 40 and prove that the smallest length of threefold rotationally symmetric trefoils is 42. We construct explicit galleries that knot as 9_35, 9_40, 9_41 and 9_47 in a way that has threefold rotational symmetry. We explain the construction of these galleries for 9_47 carefully. We conclude with three questions inspired by this work.

Knots and Coxeter Groups

TL;DR

This work investigates knot construction within the affine Coxeter complex of type by using threefold rotational symmetry. It introduces and analyzes length invariants , , and , proving and that the smallest threefold-symmetric knot length is , realized by a symmetric trefoil with . By translating cubic-lattice pieces into galleries through bar-corner gluings and frame methods, the authors construct explicit order- galleries for knots , providing concrete bounds on their threefold-symmetric knot lengths and detailing the translation between lattices. The paper culminates with a sequence of constructive techniques and three open questions about triviality criteria, symmetry-minima interactions, and whether the trefoil is always the minimal knot under symmetric stick constraints. These results deepen understanding of knotting within Coxeter lattices and offer algorithmic tools for generating symmetric knotted galleries.

Abstract

In this paper we study knots created by galleries in the affine Coxeter complex of type \widewedge{B3}. We bound the stick number by 40 and prove that the smallest length of threefold rotationally symmetric trefoils is 42. We construct explicit galleries that knot as 9_35, 9_40, 9_41 and 9_47 in a way that has threefold rotational symmetry. We explain the construction of these galleries for 9_47 carefully. We conclude with three questions inspired by this work.

Paper Structure

This paper contains 18 sections, 18 theorems, 38 equations, 29 figures, 2 tables.

Table of Contents

  1. Motivation and Results
  2. Irreducible Coxeter Complexes of rank $4$
  3. Construction of the tesselation and galleries
  4. Finite orders
  5. Knots and Symmetries
  6. Knotting Length in the Coxeter Lattices
  7. Translation from the Cubes to the Pyramids
  8. Codifying the cubic galleries in the static reference frame
  9. Codifying the cubic galleries in dynamic reference frame
  10. Translating from $C$ to the $\widetilde{B}_3$
  11. Comparing lattices
  12. Order Three Galleries
  13. Example of the construction for $9_{47}$
  14. Prime knots of $9$ crossings with threefold symmetry
  15. Questions
  16. ...and 3 more sections

Key Result

Theorem A

In the Coxeter Tesselation $\widetilde{B}_3$ the minimal stick number is at most $40$.

Figures (29)

  • Figure 1: The piece for $12_{503}.$
  • Figure 2: The knot $12_{503}$ built out of three pieces rotated by $120$ degrees
  • Figure 4: How a sphere looks close to a (green) vertex and the substituting path on the surface.
  • Figure 5: $CABDACB$
  • Figure 6: $CBDCDCDCBADCBA$
  • ...and 24 more figures

Theorems & Definitions (54)

  • Theorem A
  • Theorem B
  • Definition A
  • Theorem C
  • Definition 1
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Proposition 2
  • ...and 44 more