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Mixed tête-à-tête twists as monodromies associated with holomorphic function germs


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.

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Correspondence to Pablo Portilla Cuadrado.

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P. Portilla: supported by SVP-2013-067644 Severo Ochoa FPI grant and by project by MTM2013-45710-C2-2-P, the two of them by the Spanish Ministry of Economy and Competitiveness MINECO; also supported by the project ERCEA 615655 NMST Consolidator Grant.

B. Sigurðsson is supported by the ERCEA 615655 NMST Consolidator Grant, by the Basque Government through the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323.

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Portilla Cuadrado, P., Sigurðsson, B. Mixed tête-à-tête twists as monodromies associated with holomorphic function germs. Geom Dedicata 210, 43–64 (2021).

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