Presentations and representations of surface singular braid monoids

TL;DR

This paper investigates the algebraic structure of surface singular braid monoids, showing limitations of their linear representations and providing new simplified presentations, while connecting to knotted surface classifications.

Contribution

It introduces new presentations for the surface singular braid monoid with fewer relations and generators, and analyzes the faithfulness of its low-dimensional representations.

Findings

Two- and three-dimensional representations are not faithful for at least two or three strands.

New simplified presentations with fewer relations and generators.

Formulations of all knotted surfaces in Yoshikawa's table using surface singular braid monoids.

Abstract

The surface singular braid monoid corresponds to marked graph diagrams of knotted surfaces in braid form. In a quest to resolve linearity problem for this monoid, we will show that if it is defined on at least two or at least three strands, then its two or respectively three dimensional representations are not faithful. We will also derive new presentations for the surface singular braid monoid, one with reduced the number of defining relations, and the other with reduced the number of its singular generators. We include surface singular braid formulations of all knotted surfaces in Yoshikawa's table.