On the topological type of a set of plane valuations with symmetries

Let { C i : i = 1 , … , r } be a set of irreducible plane curve singularities. For an action of a finite group G , letΔ L ( { t a i } )be the Alexander polynomial in r | G | variables of the algebraic link( ⋃ i = 1 r ⋃ a ∈ Ga C i ) ∩ S ε 3and let ζ ( t 1 , … , t r ) = Δ L ( t 1 , … , t 1 , t 2 , … , t 2 , … , t r , … , t r )with | G | identical variables in each group. (If r = 1 , ζ ( t ) is the monodromy zeta function of the function germ∏ a ∈ Ga ∗ f , where f = 0 is an equation defining the curve C 1 .) We prove that ζ ( t 1 , … , t r ) determines the topological type of the link L . We prove an analogous statement for plane divisorial valuations formulated in terms of the Poincaré series of a set of valuations.