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 (2p,q)–torus or (2p,q)–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
References
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