Isolated Critical Points of Maps from R4 to R2 and a Natural Splitting of the Milnor Number of a Classical Fibred Link, Part II

Let K be a fibred link in a 3-sphere in C², (K,ϕ_K,ψ_K) an open-book structure on the sphere. There is a field S_K of (not everywhere tangent) oriented real 2-planes on the sphere, built from (K,ϕ_K,ψ_K); homotopically S_K gives integers (λ(K),ρ(K)) which depend only on K. Here are some facts about λ(K) and ρ(K), proved in Part I of this paper [12]: the Milnor number μ(K) of K (rank of the first homology of any fibre surface F_t = ψ⁻¹(exp it)) equals λ(K) + ρ(K); if (K,ϕ_K,ψ_K) is a braided open-book structure on the (smoothed) bidisk boundary, then the set pos(K) (resp., neg(K)) of points Q in E(K), at which the oriented tangent 2-plane to the fibre surface is a complex line with its complex (resp., conjugate-complex) orientation, is an oriented 1-cycle in E(K); and λ(K) = ℓk(K,neg(K)) + ℓk(pos(K),neg(K)), ρ(K) = λ(Rev K) (where Rev K is the mirror image of K). (See §1.)