Über die Verzweigung einer algebraischen Funktion zweier Veränderlichen in der Umgebung einer singulären Stelle

The Algebraic Fundamental Group of a Curve Singularity

“…The generators and relations in our theorem (Theorem 0.0 below) for the prime to p part of the algebraic fundamental group coincide with those of Brauner's theorem [3,10,15]. The prime to p part of a fundamental group π 1 is the quotient π p 1 of π 1 by the closed normal subgroup generated by its p-Sylow subgroups.…”

“…A classical theorem of Brauner [3] (also Kähler [10], Zariski [15]) gives a formula for the generators and relations of the topological fundamental group of the knot determined by the germ of an analytically irreducible singular curve in C 2 . These elegant formulas depend only on the characteristic pairs of a Puiseux series expansion of the curve.…”

Cabling in the Skyrme–Faddeev model

“…To show that an expansion generates a particular knot we resort to an approach first used by Kähler [22] and consider instead C ∩ D ξ , where D ξ = {(x, y) ∈ C 2 : |x| = ξ, |y| ≤ ξ}. It can be shown [23] that a sufficient condition for C ∩ D ξ to be isomorphic to C ∩ S 3 ξ is for the vector (x, y) to be nowhere orthogonal to all the tangent vectors on C restricted to the four ball of radius ξ with the origin removed, B 4 ξ \{(0, 0)}.…”

The monodromy of reducible plane curves

“…This construction is completely determined by the charac- [3,9,15] relates the geometry of polynomial functions of two complex variables to algebra, proving that algebraic knots and each branch of an algebraic link are iterated tube knots, where each iteration is via an ordinary torus knot. More specifically, the structure of each branch and the linking among the various branches is completely specified by the characteristic Puiseux expansion [1,10], which, for a fixed branch, expresses y as a fractional power series in x.…”

“…The problem is most amenable to solution in the case n= 1, because the geometry is completely understood [3,9,15]. In the classical case (n=l), f-l(0) is called a plane curve, and the algebraic link L 1 has r components, one corresponding to each of the branches of f-1(0) at the origin.…”