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Bounds in simple hexagonal lattice and classification of 11-stick knots

Yueheng Bao, Ari Benveniste, Marion Campisi, Nicholas Cazet, Ansel Goh, Jiantong Liu, Ethan Sherman

Abstract

The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial 11-stick knots in the sh-lattice are the trefoil knot ($3_1$) and the figure-eight knot ($4_1$).

Bounds in simple hexagonal lattice and classification of 11-stick knots

Abstract

The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally, we show that the only non-trivial 11-stick knots in the sh-lattice are the trefoil knot () and the figure-eight knot ().
Paper Structure (6 sections, 30 theorems, 10 equations, 13 figures)

This paper contains 6 sections, 30 theorems, 10 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Preliminary Definitions
  3. Linear Transformation between Lattices
  4. Upper Bound of Stick Number and Edge Length in sh-lattice
  5. Classification of 11-stick Knots
  6. Future Work

Key Result

Theorem 1

For any knot type $[K]$, $s_{sh}[K] < s_{L}[K]$, where $s_L$ and $s_{sh}$ are the stick numbers of $[K]$ in the cubic lattice and in the simple hexagonal lattice, respectively.

Figures (13)

  • Figure 1: Effect of $T$ on the trefoil knot
  • Figure 2: Transforming an $xy$-corner to a $z$-stick
  • Figure 3: Illustration of scaling
  • Figure 4: Reducing edge length by 1
  • Figure 5: $4_1$ knot in sh-lattice with 11 sticks
  • ...and 8 more figures

Theorems & Definitions (67)

  • Theorem
  • Theorem
  • Theorem
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • proof
  • Definition 4.1: Stick Number of a Knot Type
  • ...and 57 more