Bounds on the mosaic number of Legendrian Knots

TL;DR

This paper establishes bounds on the mosaic number of Legendrian knots using classical invariants, provides examples demonstrating bound sharpness, and offers algorithms and computational results for classifying Legendrian knot mosaics.

Contribution

It introduces lower bounds on the mosaic number of Legendrian knots, constructs upper bounds for unknots, and develops an algorithm for counting Legendrian link mosaics.

Findings

Lower bounds relate mosaic number to classical invariants.

Examples show bounds are sharp in certain cases.

Updated census of Legendrian knots with mosaic number ≤6.

Abstract

Mosaic tiles were first introduced by Lomonaco and Kauffman in 2008 to describe quantum knots, and have since been studied for their own right. Using a modified set of tiles, front projections of Legendrian knots can be built from mosaics as well. In this work, we compute lower bounds on the mosaic number of Legendrian knots in terms of their classical invariants. We also provide a class of examples that imply sharpness of these bounds in certain cases. An additional construction of Legendrian unknots provides an upper bound on the mosaic number of Legendrian unknots. We also adapt a result of Oh, Hong, Lee, and Lee to give an algorithm to compute the number of Legendrian link mosaics of any given size. Finally, we use a computer search to provide an updated census of known mosaic numbers for Legendrian knots, including all Legendrian knots whose mosaic number is 6 or less.