  We investigate some algebraic structures called quasi-trivial quandles and we
use them to study link-homotopy of pretzel links. Precisely, a necessary and
sufficient condition for a pretzel link with at least two components being
trivial under link-homotopy is given. We also generalize the quasi-trivial
quandle idea to the case of biquandles and consider enhancement of the
quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle
cocycles, obtaining invariants of link-homotopy type of links analogous to the
quasi-trivial quandle cocycle invariants in Ayumu Inoue's article
arXiv:1205.5891.
