Geography of Legendrian knot mosaics

Abstract

In this paper, mosaic tiles have been used to build knots and define invariants since they were introduced by Lomonaco and Kauffman to describe quantum knots. In this paper, we propose a modified set of tiles to describe the front projections of Legendrian knots. We explore the effect of stabilization on the mosaic number of a Legendrian knot by defining and classifying “two-cell” stabilizations. Inspired by a construction of Ludwig et al. in 2013, we provide an infinite family of Legendrian unknots whose mosaic numbers are realized only in non-reduced projections. Finally, we provide a census of known mosaic numbers for Legendrian unknots and trefoils.