Positive links are strongly quasipositive

Abstract

Given an oriented link diagram D, let X>(D) (resp., X<(D)) be the set of positive (resp., negative) crossings, and O≥(D) (resp., O<(D)) the set of Seifert circles adjacent to some x ∈ X>(D) (resp., to no x ∈ X>(D)). Write #A for the number of elements of A. Let S(D) be the surface produced by applying Seifert’s algorithm to D; let L(D): = ∂S(D). An oriented link L is positive if L is isotopic to L(D) for some D with #X<(D) = 0, almost positive if L is not positive but L is isotopic to L(D) for some D with #X<(D) = 1, and strongly quasipositive if L bounds a quasipositive Seifert surface, that is, a surface embedded incompressibly on a fiber surface plumbed from positive Hopf annuli. Theorem. If #X<(D) = 0, then S(D) is quasipositive, and so L(D) is strongly quasipositive. Denote by χs(L) the greatest value of the euler characteristic χ(F) for F ⊂ D 4 a smooth, oriented 2-manifold such that L = ∂F and F has no closed components. Let D 1 (K) be the untwisted positive Whitehead double of a knot K, D k (K): = D 1 (D k−1 (K)).