  We offer a pedestrian level review of the wall-crossing invariants. The story
begins from the scattering theory in quantum mechanics where the spectrum
reshuffling can be related to permutations of S-matrices. In non-trivial
situations, starting from spin chains and matrix models, the S-matrices are
operator-valued and their algebra is described in terms of R- and mixing
(Racah) U-matrices. Then, the Kontsevich-Soibelman invariants are nothing but
the standard knot invariants made out of these data within the
Reshetikhin-Turaev-Witten approach. The R- and Racah matrices acquire a
relatively universal form in the quasiclassical limit, where the basic
reshufflings with the change of moduli are those of the Stokes line. Natural
from this point of view are matrices provided by the modular transformations of
conformal blocks (with the usual identification R=T and U=S), and in the
simplest case of the first degenerate field (2,1), when the conformal blocks
satisfy a second order Shroedinger-like equation, the invariants coincide with
the Jones (N=2) invariants of the associated knots. Another possibility to
construct knot invariants is to realize the cluster coordinates associated with
reshufflings of the Stokes lines immediately in terms of check-operators acting
on the solutions to the Knizhnik-Zamolodchikov equations. Then, the R-matrices
are realized as products of successive mutations in the cluster algebra and are
manifestly described in terms of quantum dilogarithms ultimately leading to the
Hikami construction of knot invariants.
