Immersed M\"obius bands in knot complements and representatives of $\mathbb{Z}_2$-homology classes

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 nonntrivial $(2p,q)$-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. We then use these constructions to find, for any $n\geq 2$, an oriented 3-manifold $Y_n$ and class $\alpha_n \in H_2(Y_n;\mathbb{Z}_2)$ such that $\alpha_n$ can be represented by an immersed $\mathbb{RP}^2$, but any embedded representative of $\alpha_n$ has a component $S$ with $\chi(S) \leq 1-n$.