On Meta-monoids and the Fox-Milnor Condition

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 $\mathrm{\Gamma }$-calculus that gives an Alexander invariant of tangles. We believe that $\mathrm{\Gamma }$-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.