In http://www.math.msu.edu/~akbulut/papers/akbulut.lec.pdf, which is a (still developed) set of lecture notes on 4-manifolds by Selman Akbulut, in section 1.5 there is a way to draw a non-orientable handle.

When reading this, I thought, this is just as cumbersome as drawing a 1-handle as a pair of balls (we now have the dotted circle for that, although (as Akbulut says) both notations have their advantages). For example when you have more than one nonoriented 1-handle.

My question is:

  • Is there another, invariant way to depict nonorientation in a Kirby diagram ?
  • As I understand it sofar, the notation used by Akbulut is mainly used for 1-handles. Is there a way to enhance Kirby diagrams such that nonorientation of 2-handles is also depicted ?
  • $\begingroup$ I think 2-handles can't be nonoriented in a reasonable way. $\endgroup$ – Turion May 5 '16 at 19:10
  • $\begingroup$ As for your first question, I'd guess that you can still use Akbulut's carving notation, but you need to put a special marking on the 1-handle and take care that all the framing coefficients of 2-handles going through it are elements in $\mathbb{Z}_2$, not $\mathbb{Z}$. $\endgroup$ – Turion Jul 18 '17 at 10:50

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