# Given a Heegaard splitting $M = V\cup_F W$, then $V\setminus N(D_1)$ is ambient isotopic to $V\cup N(D_2)$ for a meridian pair $\{D_1,D_2\}$

I sincerely apologize if MathOverflow is not the appropriate place to ask this question. I also tried consulting M.SE but it seems that this question gained little to no interest .

Consider a Heegaard splitting $$M = V\cup_F W$$ of a $$3$$-manifold $$M$$ with splitting surface $$F = \partial V = \partial W$$. Suppose there is a meridian pair $$\{D_1,D_2\}$$, that is, $$D_1$$ is a meridian disk in $$V$$ and $$D_2$$ a meridian disk in $$W$$. Both disks intersect transversaly so that their boundary curves $$\partial D_1$$ and $$\partial D_2$$ intersect in exactly one point.

Now I would like to verify that $$V\setminus N(D_1)$$ is ambient isotopic to $$V\cup N(D_2)$$ where $$N(\cdot)$$ denotes a regular neighbourhood. But I'm somewhat new to geometric topology and I am not sure on how to proceed.

## My attempt:

What I know is that both disks $$D_i$$ are compression disks in $$V$$ respectively $$W$$, thus we can compress along $$D_1$$ in $$V$$, that is, we cut at the regular neighbourhood $$N(D_1)$$ of $$D_1$$ in order to obtain $$V\setminus N(D_1)$$.

Now the union of $$N(D_1)$$ and $$N(D_2)$$ along the square in which they intersect is a $$3$$-ball and so we can isotope "through" that ball.

But now I'm unsure if this is the correct attempt and if this already shows that $$V\setminus N(D_1)$$ is ambient isotopic to $$V\cup N(D_2)$$.

Another Idea was to somehow construct the deired isotopy in a "pedestrian way" by considering a collar of the boundary $$\partial V$$ and somehow extending an isotopy from the boundary to the entire handlebody $$V$$ but so far I haven't had much success with this approach either.

Any feedback would be highly appreciated! Thanks in advance.

Yes, you are correct that the two handlebodies ($$V$$ cut along $$D$$ and $$V$$ plus a neighbourhood of $$E$$) are isotopic. This is discussed when defining stabilisation and the inverse operation destabilisation of Heegaard splittings. I don’t have a copy of Hempel’s book with me, but I am fairly sure it is discussed there.