An infinite family of knots whose hexagonal mosaic number is only realized in non-reduced projections

TL;DR

This paper introduces an infinite family of knots that can be embedded in hexagonal mosaics of a certain size but only in non-reduced projections, extending previous rectangular mosaic results.

Contribution

It presents a new family of knots with specific mosaic embedding properties and introduces a tool for systematically finding all flypes in link diagrams.

Findings

Identifies knots with embedding constraints in hexagonal mosaics.

Extends previous mosaic results from rectangular to hexagonal grids.

Provides a method to find all minimal crossing embeddings of prime, alternating knots.

Abstract

We give an infinite family of knots such that for any given $r \geq 3$, the family contains a knot which can be embedded on a hexagonal $r$-mosaic, but cannot fit on a hexagonal $r$-mosaic in an embedding that achieves its crossing number. This extends the rectangular mosaic result of Ludwig, Evans, and Paat. We also introduce a new tool for systematically finding all possible flypes for the diagram of any link thus making it easier to find all possible minimal crossing embeddings of prime, alternating knots.