#### Volume 20, issue 2 (2020)

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Immersed Möbius bands in knot complements

### Mark C Hughes and Seungwon Kim

Algebraic & Geometric Topology 20 (2020) 1059–1072
##### Abstract

We study the $3$–dimensional immersed crosscap number of a knot, which is a nonorientable analogue of the immersed Seifert genus. We study knots with immersed crosscap number $1$, and show that a knot has immersed crosscap number $1$ if and only if it is a nontrivial $\left(2p,q\right)$–torus or $\left(2p,q\right)$–cable knot. We show that unlike in the orientable case the immersed crosscap number can differ from the embedded crosscap number by arbitrarily large amounts, and that it is neither bounded below nor above by the $4$–dimensional crosscap number.

##### Keywords
knots, Möbius bands, immersed surfaces
##### Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Secondary: 57M35
##### Publication
Received: 15 January 2019
Revised: 3 September 2019
Accepted: 16 January 2020
Published: 23 April 2020
##### Authors
 Mark C Hughes Department of Mathematics Brigham Young University Provo, UT United States Seungwon Kim Center for Geometry and Physics Institute for Basic Science (IBS) Pohang South Korea