  In this paper, we study alternating links in thickened surfaces in terms of
the lattices of integer flows on their Tait graphs. We use this approach to
give a short proof of the first two generalised Tait conjectures. We also prove
that the flow lattice is an invariant of alternating links in thickened
surfaces and is further invariant under disc mutation. For classical links, the
flow lattice and $d$-invariants are complete invariants of the mutation class
of an alternating link. For links in thickened surfaces, we show that this is
no longer the case by finding a stronger mutation invariant, namely the
Gordon-Litherland linking form. In particular, we find alternating knots in
thickened surfaces which have isometric flow lattices but with non-isomorphic
linking forms.
