The Dynkin-presentation conjecture for discriminant-complement groups

From papers

Let ff be a singular function with Milnor number 4mu4mu, and let a Dynkin diagram of ff have vertex cardinality 4mu4mu. Write t1,,t4mut_1,\cdots,t_4mu for generators indexed by its vertices, and let II denote the vertex index set. Dynkin-presentation conjecture. The fundamental group of the discriminant complement of a versal unfolding of ff has the presentation

ti,1cleqicleq4mu | titjti=tjtitj,i<j joined by an edge of weight pm1,titj=tjti,i<j not joined by an edge,titjtkti=tjtktitj,i<j<k in an edge triangle of weight product 1.\left\langle t_i,\,1cleq icleq4mu\ \middle|\ \begin{array}{ll} t_i t_j t_i=t_j t_i t_j,& i<j\text{ joined by an edge of weight }pm1,\\ t_i t_j=t_j t_i,& i<j\text{ not joined by an edge},\\ t_i t_j t_k t_i=t_j t_k t_i t_j,& i<j<k\text{ in an edge triangle of weight product }-1 \end{array}\right\rangle.

Here the relations range over the indicated vertices in II. This is proposed as a preferred finite presentation and would determine the fundamental group from the Dynkin diagram; the source gives no resolution.

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Sources & referencesView supporting material

Primary source

Michael Lönne, “Braid Monodromy of Hypersurface Singularities”, arXiv:math/0602371 (2006).

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