  In this paper we introduce an algebraic structure known as meta-monoids which
is particularly suited for the study of knot theory. We define a meta-monoid
called $\Gamma$-calculus that gives an Alexander invariant of tangles. We
believe that $\Gamma$-calculus gives a unifying framework to study the
Alexander polynomial. Specifically, we rederive certain important properties of
the Alexander polynomial, most notably the Fox-Milnor condition on the
Alexander polynomials of ribbon knots. We argue that our proof has some
potential for generalization which may help tackle the slice-ribbon conjecture.
