  Let $C$ be an algebraic curve defined by a sufficiently generic bivariate
Laurent polynomial with given Newton polygon $\Delta$. It is classical that the
geometric genus of $C$ equals the number of lattice points in the interior of
$\Delta$. In this paper we give similar combinatorial interpretations for the
gonality, the Clifford index and the Clifford dimension, by removing a
technical assumption from a recent result of Kawaguchi. More generally, the
method shows that apart from certain well-understood exceptions, every
base-point free pencil whose degree equals or slightly exceeds the gonality is
'combinatorial', in the sense that it corresponds to projecting $C$ along a
lattice direction. We then give an interpretation for the scrollar invariants
associated to a combinatorial pencil, and show how one can tell whether the
pencil is complete or not. Among the applications, we find that every smooth
projective curve admits at most one Weierstrass semi-group of embedding
dimension $2$, and that if a non-hyperelliptic smooth projective curve $C$ of
genus $g \geq 2$ can be embedded in the $n$th Hirzebruch surface
$\mathcal{H}_n$, then $n$ is actually an invariant of $C$.
