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Bounding Crossing Number in Rectangular and Hexagonal Knot Mosaics

Hugh Howards, Jiong Li, Xiaotian Liu

Abstract

Howards and Kobin give a sharp upper bound for crossing number of knots on rectangular mosaics. Here we extend the proof to create a new bound for hexagonal mosaics in all three natural settings and shorten the proof in the rectangular setting.

Bounding Crossing Number in Rectangular and Hexagonal Knot Mosaics

Abstract

Howards and Kobin give a sharp upper bound for crossing number of knots on rectangular mosaics. Here we extend the proof to create a new bound for hexagonal mosaics in all three natural settings and shorten the proof in the rectangular setting.

Paper Structure

This paper contains 6 sections, 10 theorems, 6 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Families of links denoted $L_r$ and a families of knots denoted $A_r$
  4. Useful Lemmas
  5. Bounding Crossing Number
  6. Open Questions And Conjectures

Key Result

Lemma 4.1

Let $\ddddot{M_1}$ and $\ddddot{M_2}$ be saturated standard hexagonal $r$-mosaics. A rotation of $\ddddot{M_1}$ by 0 or $\frac{\pi}{3}$ together with a combination of crossing changes and planar isotopies of individual tiles is sufficient to get from $\ddddot{M_1}$ to $\ddddot{M_2}$.

Figures (18)

  • Figure 1.1: Here we see the hexagonal tiles up to rotation. Numbers to the left of the tile are given so that we can refer to specific tiles by name when convenient.
  • Figure 1.2: On the left we see a standard mosaic consisting of the connect sum of 23 trefoils built using only Tiles 27, 17, and 5 on the interior. On the far right we see an enhanced mosaic, but it also shows how to lower the crossing number in any of the settings by multiples of 3 by replacing Tile 27 by Tile 17. The center and right figures show that in the enhanced setting we can add in our choice of one or two more crossings in the boundary tiles in the enhanced setting to achieve crossing numbers of the form $3k+1$ and $3k+2$.
  • Figure 2.1: Here we see three ways of connecting up a collection of interior tiles through the boundary. The interior tiles on the three mosaics are identical and only the boundary tiles are different. The link on the left is $\ddddot{L_4}$ (a standard hexagonal mosaic). The link on the right is $\widehat{L_4}$ (an enhanced hexagonal mosaic).
  • Figure 2.2: The trefoil embedded in a hexagonal mosaic with $r=2$. We note that for $r=2$ standard mosaics and enhanced mosaics are the same because all boundary tiles are corner tiles so it is impossible to have crossings in the boundary tiles. This single mosaic can be thought of as $\ddddot{L_2}$, $\hat{L_2}$, $\widehat{L_2}$, $\ddddot{A_2}$, $\hat{A_2}$, and $\widehat{A_2}$.
  • Figure 2.3: The figure shows the knot intersecting a square tile in green arcs and the complement is drawn in blue. In hk the complement for Tile 5 was blank, but Tile 4 had a single dot on it and was called a type 0 tile. The type 0 tile is not needed in this paper and for both Tiles 4 and 5 the complement is trivial since both of those tiles contain 2 arcs of the link hitting all 4 possible connection points for the square tile.
  • ...and 13 more figures

Theorems & Definitions (26)

  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Claim 4.4
  • proof
  • Claim 4.5
  • Lemma 4.6
  • ...and 16 more