  We study autoequivalences of $D^{b}Coh(X)$ associated to B-brane transport
around loops in the stringy K\"ahler moduli of $X$. We consider the case of $X$
being certain resolutions of determinantal varieties embedded in
$\mathbb{P}^{d}\times G(k,n)$. Such resolutions have been modeled, in general,
by nonabelian gauged linear sigma models (GLSM). We use the GLSM construction
to determine the window categories associated with B-brane transport between
different geometric phases using the machinery of grade restriction rule and
the hemisphere partition function. In the family of examples analyzed the
monodromies around phase boundaries enjoy the interpretation as loop inside
link complements. We exploit this interpretation to find a decomposition of
autoequivalences into simpler spherical functors and we illustrate this in two
examples of Calabi-Yau 3-folds $X$, modeled by an abelian and nonabelian GLSM
respectively. In additon we also determine explicitly the action of the
autoequivalences on the Grothendieck group $K(X)$ (or equivalently, B-brane
charges).
