Constructions of and Bounds on the Toric Mosaic Number

TL;DR

This work introduces toric mosaics by identifying opposite edges of classical knot mosaics, defining the toric mosaic number $m_T(K)$ and establishing key bounds such as $m_T(K)\le m(K)$ with $m_T(U)=1$ and $m_T(K)=2$ iff $K$ is the trefoil. It develops two constructive algorithms—the one-braid and full-braid methods—to bound $m_T(K)$ for $(p,q)$-torus knots, providing concrete bounds like $m_T(K)\le (q+1)/2$ for $(2,q)$ and $m_T(K)\le 2n$ in a family of $(2,q')$-torus knots, with improvements for odd $q$ up to $q'$. The paper also reports a computer census using base-11 encodings and HOMFLY-PT computations to realize toric mosaics of small size, identifying nine prime knots on toric $3$-mosaics and showing all knots up to crossing number $9$ occur on toric $4$-mosaics. The results offer a framework for efficiently constructing toric mosaics, raise questions about generalizing to broader torus embeddings and other surfaces, and hint at connections to virtual knot mosaics and broader knot-theoretic invariants.

Abstract

Knot mosaics were introduced by Kauffman and Lomonaco in the context of quantum knots, but have since been studied for their own right. A classical knot mosaic is formed on a square grid. In this work, we identify opposite edges of the square to form mosaics on the surface of a torus. We provide two algorithms for efficiently constructing toric mosaics of torus knots, providing upper bounds for the toric mosaic number. Using these results and a computer search, we provide a census of known toric mosaic numbers.

Figures (11)

• Figure 1: The mosaic tiles $T_0,\ldots,T_{10}$.
• Figure 2: A $4$-mosaic representing a trefoil knot.
• Figure 3: A toric $2$-mosaic representing a trefoil knot.
• Figure 4: The classical representation of a $(3,4)$-torus knot and its eight irreducible hidden crossings.
• Figure 5: A toric $4$-mosaic representing the $(3,4)$-torus knot inspired by its classical representation.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Proposition 1
• Proposition 2
• Theorem 1
• Corollary 1
• Corollary 2
• Definition 3
• Definition 4
• proof : Proof of Theorem \ref{['thm:rapunzel']}