From Lickorish-Wallace theorem, every 3-manifold is an integral surgery on a link in $S^3$. From its proof from Saveliev's book, it seems obvious that if I know the Heegaard splitting of a closed 3-manifold $M$, I can get a link on which performing surgery gives $M$. Now I want to get a surgery link from a given explicit Heegaard splitting of a manifold. Is there some known easy way for this process?



Getting surgery link from Heegaard splitting
Yes- it's easy. There's the "digging the trench" construction, nicely described in [A simple proof of the fundamental theorem of Kirby calculus on links](http://www.ams.org/journals/tran/1992-331-01/S0002-9947-1992-1065603-2/S0002-9947-1992-1065603-2.pdf) by Ning Lu, for example.


In short and with all details suppressed, take the curve around which you Dehn-twist on the Heegaard surface, dig a little trench under it in one of the handlebodies, and take that trench to be the tubular neighbourhood of your surgery component. The framing is either plus of minus one, depending on which way you are Dehn twisting.


