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Akbulut's cork is the Mazur manifold $W$ shown in the picture below,enter image description here

This manifold carries an involution of it's boundary $f:\partial W\to \partial W$ that exchanges a meridian of the 0-framed curve with a meridian of the dotted curve.

In Akbulut - 4-manifolds, the author says that $f$ is related to performing surgeries to exchange the dot with the 0-framed knot. I don't understand how this relates to $f$ and how $f$ is defined.

We can surely perform two surgeries (one along a circle and one along a 2-sphere) to get a new 4-manifold $\tilde{W}$ that has a Kirby diagram obtained by exchanging dot and zero in the above picture. However surgery does not affect the boundary so I do not see how this could induce a nontrivial map between the boundaries.

Precisely how is $f$ defined?

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The boundary of W may be described as 0-framed surgery on both components of the link you drew. The link can be drawn in a more symmetric fashion, so that it is clear that there is an involution interchanging the two components. See for instance Figure 4 of Akbulut's paper, A solution to a conjecture of Zeeman, Topology 30 (1991, 513-515. This involution induces an involution on the surgered manifold that interchanges the two meridians.

If you don't care that the diffeomorphism on the boundary is an involution, you can just do an isotopy interchanging the components.

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