If $\vec{n} = (n_1,...,n_k)$ is a vector of integers, there seems to be a well-defined homomorphism

$B_k \ltimes \left(B_{n_1} \times \cdots \times B_{n_k}\right) \to B_N$

where $N = \sum n_i$ and $B_N$ denotes the braid group with $N$ strands. To see this, use the $k$ braids supplied by the right factors as strands for the "large" braid specified by the left factor $B_k$. Let us call a braid $b \in B_N$ *nested* if it lies in the image of such a homomorphism for some $\vec n \neq (1,...,1), (N)$.

Are there algorithms for determining whether a braid is nested?

Thanks!

**EDIT:** Here nested braids are called "reducible" following the Nielsen-Thurston classification. Not sure whether the classification is effective.

As has been pointed out in the comments, the question can also be reformulated in terms of the composition map of the little squares operad.