Crossing Number Bound in Knot Mosaics

TL;DR

This paper establishes an upper bound on the crossing number of a knot based on its mosaic number, providing a mathematical link between knot complexity and mosaic representation.

Contribution

It introduces a new upper bound for the crossing number of knots in terms of their mosaic number, advancing the mathematical understanding of knot mosaics.

Findings

Crossing number c is at most (m - 2)^2 - 2 for odd m.

Crossing number c is at most (m - 2)^2 - (m - 3) for even m.

Provides a mathematical bound linking knot complexity and mosaic size.

Abstract

Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer $m$ such that the knot can be represented as a knot $m$-mosaic. In this paper we establish an upper bound for the crossing number of a knot in terms of the mosaic number. Given an $m$-mosaic and any knot $K$ that is represented on the mosaic, its crossing number $c$ is bounded above by $(m - 2)^{2} - 2$ if $m$ is odd, and $(m - 2)^{2} - (m - 3)$ if $m$ is even.