Mixed tête-à-tête twists as monodromies associated with holomorphic function germs
Abstract
Tête-à-tête graphs were introduced by N. A’Campo in 2010 with the goal of
modeling the monodromy of isolated plane curves. Mixed tête-à-tête graphs provide a
generalization which define mixed tête-à-tête twists, which are pseudo-periodic automorphisms
on surfaces. We characterize the mixed tête-à-tête twists as those pseudo-periodic
automorphisms that have a power which is a product of right-handed Dehn twists around
disjoint simple closed curves, including all boundary components. It follows that the class
of tête-à-tête twists coincides with that of monodromies associated with reduced function
germs on isolated complex surface singularities. Finally, using the language of plumbing
calculus, we relate horizontal open book decompositions of graph manifolds with mixed
tête-à-tête graphs via two algorithms.