Singular braids, singular links and subgroups of camomile type

Abstract

In this paper we find a finite set of generators and defining relations for the singular pure braid group $S{P}_n$, $n\ge 3$, that is a subgroup of the singular braid group $S{G}_n$. Using this presentation, we prove that the center of $S{G}_n$ (which is equal to the center of $S{P}_n$ for $n\ge 3$) is a direct factor in $S{P}_n$ but it is not a direct factor in $S{P}_n$. We introduce subgroups of camomile type and prove that the singular pure braid group $S{P}_n$, $n\ge 5$, is a subgroup of camomile type in $S{G}_n$. 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.

23 pages, 4 figure