Communications in Analysis and Geometry

Volume 30 (2022)

Number 5

Double branched covers of knotoids

Pages: 1007 – 1057



Agnese Barbensi (Mathematical Institute, University of Oxford, United Kingdom)

Dorothy Buck (Department of Mathematical Sciences, University of Bath, United Kingdom; and Math Department, Duke University, Durham, North Carolina, U.S.A.)

Heather A. Harrington (Mathematical Institute, University of Oxford, United Kingdom)

Marc Lackenby (Mathematical Institute, University of Oxford, United Kingdom)


By using double branched covers, we prove that there is a $1\textrm{-}1$ correspondence between the set of knotoids in $S^2$, up to orientation reversion and rotation, and knots with a strong inversion, up to conjugacy. This correspondence allows us to study knotoids through tools and invariants coming from knot theory. In particular, concepts from geometrisation generalise to knotoids, allowing us to characterise reversibility and other properties in the hyperbolic case. Moreover, with our construction we are able to detect both the trivial knotoid in $S^2$ and the trivial knotoid in $D^2$.

Received 27 February 2019

Accepted 8 October 2019

Published 17 March 2023