  In a previous paper, the second author defined integer-valued functions
delta_n on the first cohomology of a 3-manifold, generalizing McMullen's
Alexander norm. It was shown that these functions give lower bounds on the
Thurston norm. In this paper, we reformulate these invariants in terms of
Reidemeister torsion over a non-commutative multivariable Laurent polynomial
ring. This allows us to show that these functions are semi-norms.
