Rectangular mosaics for virtual knots

TL;DR

This paper introduces rectangular mosaics for virtual knots, extending previous mosaic knot models to rectangular grids with boundary identifications, and develops invariants and algorithms for virtual knot analysis.

Contribution

It defines rectangular virtual knot mosaics, adapts mosaic moves to this setting, and provides computational tools for virtual knot invariants.

Findings

Defined rectangular mosaics for virtual knots.

Modified mosaic moves for the rectangular setting.

Provided algorithms for virtual knot invariant computation.

Abstract

Mosaic knots, first introduced in 2008 by Lomanoco and Kauffman, have become a useful tool for studying combinatorial invariants of knots and links. In 2020, by considering knot mosaics on $n \times n$ polygons with boundary edge identification, Ganzell and Henrich extended the study of mosaic knots to include virtual knots - knots embedded in thickened surfaces. They also provided a set of virtual mosaic moves preserving knot and link type. In this paper, we introduce rectangular mosaics for virtual knots, defined to be $m \times n$ arrays of classical knot mosaic tiles, along with an edge identification of the boundary of the mosaic, whose closures produce virtual knots. We modify Ganzell and Henrich's mosaic moves to the rectangular setting, provide several invariants of virtual rectangular mosaics, and give algorithms for computations of common virtual knot invariants.