On Meta-monoids and the Fox-Milnor Condition

Abstract

In this paper we introduce an algebraic structure known as meta-monoids which is suited for the study of knot theory. We define a particular meta-monoid called Γ-calculus that gives an Alexander invariant of tangles. Using the language of Γ-calculus, we rederive certain important properties of the Alexander polynomial, most notably the Fox-Milnor condition on the Alexander polynomials of ribbon knots [Lic97, FM66]. We argue that our proof has some potential for generalization which may help tackle the slice-ribbon conjecture. In a sense this paper is an extension of [BNS13].