Singular braids, singular links and subgroups of camomile type

TL;DR

This paper provides a new presentation for the singular pure braid group, explores its algebraic structure, introduces camomile type subgroups, and constructs invariants for singular links.

Contribution

It introduces a finite generating set and relations for the singular pure braid group and defines new subgroup types and invariants for singular links.

Findings

The center of the singular braid group is a direct factor in the pure subgroup.

Singular pure braid groups of at least five strands are subgroups of camomile type.

Constructed fundamental singquandle as an invariant for singular links.

Abstract

In this paper we find a finite set of generators and defining relations for the singular pure braid group $SP_n$, $n \geq 3$, that is a subgroup of the singular braid group $SG_n$. Using this presentation, we prove that the center of $SG_n$ (which is equal to the center of $SP_n$ for $n \geq 3$) is a direct factor in $SP_n$ but it is not a direct factor in $SP_n$. We introduce subgroups of camomile type and prove that the singular pure braid group $SP_n$, $n \geq 5$, is a subgroup of camomile type in $SG_n$. Also we construct the fundamental singquandle using a representation of the singular braid monoid by endomorphisms of free guandle. For any singular link we define some family of groups which are invariants of this link.