Discussion of “Permian—Triassic vertebrate footprints from South Africa: Ichnotaxonomy, producers and biostratigraphy through two major faunal crises” by Marchetti, L., Klein, H., Buchwitz, M., Ronchi, A., Smith, R.M.H., DeKlerk, W.J., Sciscio, L., and Groenewald, G.H. (2019)

Gastaldo, Robert A.; Neveling, Johann

doi:10.1016/j.gr.2019.08.003

Abstract We define and study thecontact cut graphwhich is an analogue of Hatcher and Thurston's cut graph for contact geometry, inspired by contact Heegaard splittings (Giroux,Proceedings of the International Congress of Mathematicians, Beijing, 2002; Torisu,Internat. Math. Res. Notices(2000), 441–454). We show how oriented paths in the contact cut graph correspond to Lefschetz fibrations and multisection with divides diagrams. We also give a correspondence for achiral Lefschetz fibrations. We use these correspondences to define a new invariant of Weinstein domains, theWeinstein‐invariant, that is a symplectic analogue of the Kirby–Thompson's ‐invariant of smooth 4‐manifolds. We discuss the relation of Lefschetz stabilization with the Weinstein ‐invariant. We present topological and geometric constraints of Weinstein domains with . We also give two families of examples of multisections with divides that have arbitrarily large ‐invariant.

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