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Moebius bands in B^3xS^1 and the square peg problem

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Speaker: 
Peter Feller
Affiliation: 
ETH Zürich
Date: 
Mon, 2019-06-24 16:30 - 17:30
Location: 
MPIM Lecture Hall
Parent event: 
MPIM Topology Seminar

Using an idea of Hugelmeyer, we give a knot theory reproof of the following theorem. Every smooth Jordan curve in the Euclidian plane has an inscribed square.

Our knot theory result, which allows the above application, is the following. For integers n>1 that are not squares, the torus knot T(1,2n) in S^2xS^1 does not arise as the boundary of a locally-flat Moebius band in B^3xS^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.
We will comment on how our locally flat knot theory result provides a result for Jordan curves with a weaker regularity assumption.
 
Based on work in progress with Marco Golla.

 

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