Bounding Crossing Number in Rectangular and Hexagonal Knot Mosaics

TL;DR

This paper extends existing bounds on the crossing number of knots from rectangular mosaics to hexagonal mosaics, providing new bounds and simplified proofs across different mosaic settings.

Contribution

It introduces a new upper bound for the crossing number in hexagonal mosaics and simplifies the proof for rectangular mosaics, advancing knot mosaic theory.

Findings

New upper bound for hexagonal knot mosaics crossing number

Simplified proof for rectangular mosaic crossing bounds

Extension of bounds to all three natural settings

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.