Bounds on the mosaic number of Legendrian Knots
Margaret Kipe, Samantha Pezzimenti, Leif Schaumann, Luc Ta, Wing Hong Tony Wong
TL;DR
This work establishes both lower and upper bounds for the mosaic numbers of Legendrian knots by connecting front-projection invariants to mosaic tile counts through combinatorial and linear-algebraic methods. It introduces oriented Legendrian mosaic tiles, derives two complementary lower-bound strategies, and proves sharpness in an infinite crab-bucket family. For unknots, it provides constructive upper bounds via barn and soil configurations and Kraken/fish moves, along with an explicit algorithm and infinite families attaining equality. It also develops a counting framework for Legendrian link mosaics, adapts existing counting results, and delivers an updated census of mosaic numbers up to size six (with data extended to larger sizes in appendices). The paper closes with computational censuses, examples where stabilization reduces mosaic number, and a suite of open questions aimed at extending these methods to broader knot types and invariants.
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.