  The Alexander polynomial (1928) is the first polynomial invariant of links
devised to help distinguish links up to isotopy. In recent work of the authors,
Fox's conjecture (1962) -- stating that the absolute values of the coefficients
of the Alexander polynomial for any alternating link are unimodal -- was
settled for special alternating links. The present paper is a study of the
special combinatorial and discrete geometric properties that Alexander
polynomials of special alternating links possess along with a generalization to
all Eulerian graphs, introduced by Murasugi and Stoimenow (2003). We prove that
the Murasugi and Stoimenow generalized Alexander polynomials can be expressed
in terms of volumes of root polytopes of unimodular matrices, building on the
beautiful works of Li and Postnikov (2013) and T\'othm\'er\'esz (2022). We
conjecture a generalization of Fox's conjecture to the Eulerian graph setting.
We also bijectively relate two longstanding combinatorial models for the
Alexander polynomials of special alternating links: Crowell's state model
(1959) and Kauffman's state model (1982, 2006).
