|DATE||July 13 (Fri), 2018|
|TITLE||Immersed crosscap number and Z_2 Thurston norm|
In this talk, we will talk about 3-dimensional immersed crosscap number of a knot, which is a non-orientable 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 an infinite family of 3-manifolds such that one of its second Z2 homology class can be represented by an immersed real projective plane but any embedded representative of it has a component with Euler characteristic strictly less than 1.