On a surface singular braid monoid

TL;DR

This paper introduces a monoid for knotted surfaces in four-dimensional space, establishing relations from topological moves, classifying knotted surfaces by index, and analyzing diagram unlinking complexities.

Contribution

It defines a new surface singular braid monoid with relations derived from topological moves and classifies knotted surfaces based on an index, revealing finite and infinite classes.

Findings

Exactly six knotted surface types with index ≤ 2

Infinitely many knotted surfaces with index = 3

Constructed diagrams requiring at least four Reidemeister III moves

Abstract

We introduce a monoid corresponding to knotted surfaces in four space, from its hyperbolic splitting represented by marked diagram in braid like form. It has four types of generators: two standard braid generators and two of singular type. Then we state relations on words that follows from topological Yoshikawa moves. As a direct application we will show equivalence of some known theorems about twist-spun knots. We wish then to investigate an index associated to the closure of surface singular braid monoid. Using our relations we will prove that there are exactly six types of knotted surfaces with index less or equal to two, and there are infinitely many types of knotted surfaces with index equal to three. Towards the end we will construct a family of classical diagrams such that to unlink them requires at least four Reidemeister III moves.