Immersed M\"obius bands in knot complements

2017 
We study the immersed crosscap number of a knot, which is a nonorientable analogue of the immersed Seifert genus. Unlike the orientable case, we show that 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. We then 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.
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