The Unintelligible Thread

Let K⊂S³ be an oriented knot or link, and F⊂S³ a connected oriented spanning surface for K. Let θ:H₁(F)⨯H₁→Z be the Seifert paring.

Two polynomials f(t),g(t)∈Z[t] are said to be balanced (written f≐g) if there is a non-negative integer n such that ±tⁿf(t)=g(t) or ±tⁿg(t)=f(t). This definition is also extended to rational functions. Thus t+1/t and t²+1 are balanced and we write t+1/t≐t²+1.

Let K, F, θ be as above. The Alexander polynomial A_K(t) is the balance class of the polynomial A_K(t) = D(θ-tθ'). (It follows from S-equivalence that this determinant is well-defined on isotopy classes of knots and links up to multiplication by factors of the form ±tⁿ)