On pure braids and bonded braids

Abstract

In this paper we investigate the relation between the bonded braid monoid BBn and the group of pure braids $P_n$, realised by an algebraic analogue of the topological tangle insertion, namely by replacing a bond by (a power of) a pure braid generator. To do so, we provide appropriate presentations for $P_n$ using cyclic relations and clasp generators, and extend them to presentations of $B_n$. We construct the monoid epimorphism $\Phi :B{B}_n\to Bn$, mapping bonds to pure braid generators, which is the main result of the paper. Using the isomorphism from the singular braid monoid $S{B}_n$ to the tight bonded braid monoid $B{B}_n$, we extend $\Phi$ to a new type of epimorphism $s\Phi :S{B}_n\to B_n$. The epimorphisms $\Phi$ and $s\Phi$ are extended to families of epimorphisms ${\Phi }_p$, $s{\Phi }_p$, $p\in \mathbb{Z}$.

arXiv comment45 pages, 31 figures