Crossing Number Bound in Knot Mosaics

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.

Updated to reflect referee's suggestions. 23 pages, 25 figures, to appear in Journal of Knot Theory and Its Ramifications