Moebius bands in S^1xB^3 and the square peg problem by Peter Feller

Geometry Topology Seminar
Wednesday, April 3, 2019 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 006
Peter Feller – ETH Zurich – peter.feller@math.ethz.ch
JungHwan Park

Following an idea of Hugelmeyer, we give a knot theory reproof of a theorem of Schnirelman: Every smooth Jordan curve in the Euclidian plane has an inscribed square. We will comment on possible generalizations to more general Jordan curves.

Our main knot theory result is that the torus knot T(2n,1) in S^1xS^2 does not arise as the boundary of a locally-flat Moebius band in S^1xB^3 for square-free integers n>1. For context, we note that for n>2 and the smooth setting, this result follows from a result of Batson about the non-orientable 4-genus of certain torus knots. However, we show that Batson's result does not hold in the locally flat category: the smooth and topological non-orientable 4-genus differ for the T(9,10) torus knot in S^3.

Based on joint work with Marco Golla.