On continuous families of geometric Seifert conemanifold structures

TL;DR

This paper classifies the geometric structures of certain Seifert fibered conemanifolds with limited singular fibers, providing methods to compute holonomy and applying results to specific families like spherical, Nil, and Dehn surgery manifolds.

Contribution

It determines the possible geometries of Seifert conemanifolds with up to three singular fibers and introduces a method to compute their holonomy, extending previous results to all torus knots.

Findings

Classified geometries for Seifert conemanifolds with ≤3 singular fibers.

Developed a method to obtain holonomy of these structures.

Applied results to spherical, Nil, and Dehn surgery manifolds, generalizing previous work.

Abstract

We determine the Thurston's geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space $S^2$ and no more than three exceptional fibres, whose singular set, composed by fibres, has at most 3 components which can include exceptional or general fibres (the total number of exceptional and singular fibres is less or equal than three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery in a torus knot $K_{(r,s)}$. As a consequence we generalize to all torus knots the results obtained in \cite{LM2015} for the case of the left handle trefoil knot. We associate a plot to each torus knot for the different geometries, in the spirit of Thurston.