Mosaics for immersed surface-links

TL;DR

This paper extends the concept of knot mosaics to immersed surface-links using singular marked graph diagrams, establishing a new mosaic system and defining the mosaic number for these complex structures.

Contribution

It introduces a mosaic system for immersed surface-links and defines their mosaic number, expanding the framework of knot mosaics to more complex surface-link types.

Findings

Established a mosaic system for immersed surface-links.

Defined the mosaic number for immersed surface-links.

Provided bounds and discussions on the mosaic number.

Abstract

The concept of a knot mosaic was introduced by Lomonaco and Kauffman as a means to construct a quantum knot system. The mosaic number of a given knot $K$ is defined as the minimum integer $n$ that allows the representation of $K$ on an $n \times n$ mosaic board. Building upon this, the first author and Nelson extended the knot mosaic system to encompass surface-links through the utilization of marked graph diagrams and established both lower and upper bounds for the mosaic number of the surface-links presented in Yoshikawa's table. In this paper, we establish a mosaic system for immersed surface-links by using singular marked graph diagrams. We also provide the definition and discussion on the mosaic number for immersed surface-links.