  We provide a combinatorial interpretation of the Kazhdan--Lusztig polynomial
of the matroid arising from the braid arrangement of type $\mathrm{A}_{n-1}$,
which gives an interpretation of the intersection cohomology Betti numbers of
the reciprocal plane of the braid arrangement. Moreover, we prove an
equivariant version of this result. The key combinatorial object is a class of
matroids arising from series-parallel networks. As a consequence, we prove a
conjecture of Elias, Proudfoot, and Wakefield on the top coefficient of
Kazhdan--Lusztig polynomials of braid matroids, and we provide explicit
generating functions for their Kazhdan--Lusztig and $Z$-polynomials.
