  We study the Newton polytopes of determinants of square matrices defined over
rings of twisted Laurent polynomials. We prove that such Newton polytopes are
single polytopes (rather than formal differences of two polytopes); this result
can be seen as analogous to the fact that determinants of matrices over
commutative Laurent polynomial rings are themselves polynomials, rather than
rational functions. We also exhibit a relationship between the Newton polytopes
and invertibility of the matrices over Novikov rings, thus establishing a
connection with the invariants of Bieri-Neumann-Strebel (BNS) via a theorem of
Sikorav.
  We offer several applications: we reprove Thurston's theorem on the existence
of a polytope controlling the BNS invariants of a $3$-manifold group; we extend
this result to free-by-cyclic groups, and the more general descending HNN
extensions of free groups. We also show that the BNS invariants of Poincar\'e
duality groups of type $\mathtt{F}$ in dimension $3$ and groups of deficiency
one are determined by a polytope, when the groups are assumed to be agrarian,
that is their integral group rings embed in skew-fields. The latter result
partially confirms a conjecture of Friedl.
  We also deduce the vanishing of the Newton polytopes associated to elements
of the Whitehead groups of many groups satisfying the Atiyah conjecture. We use
this to show that the $L^2$-torsion polytope of Friedl-L\"uck is invariant
under homotopy. We prove the vanishing of this polytope in the presence of
amenability, thus proving a conjecture of Friedl-L\"uck-Tillmann.
