Non-orientable slice surfaces and inscribed rectangles

Peter Feller , Marco Golla (1, 2)

1 CNRS - Centre National de la Recherche Scientifique
2 LMJL - Laboratoire de Mathématiques Jean Leray

We give a unified proof of existence of inscribed squares and inscribed rectangles with aspect ratio $\sqrt3$ in Jordan curves with a regularity condition, generalizing a result by Hugelmeyer. We also discuss differences between genera of smooth and locally-flat non-orientable surfaces in the 4-ball with boundary a given torus knot or 2-bridge knot. In particular, we establish that a result by Batson on the smooth non-orientable 4-genus of torus knots does not hold in the locally-flat category.

Domaines: Géométrie métrique [math.MG]; Topologie géométrique [math.GT]

Peter Feller, Marco Golla. Non-orientable slice surfaces and inscribed rectangles. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, In press, pp.1463-1485. ⟨10.2422/2036-2145.202105_099⟩. ⟨hal-02499749⟩