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Jul 3, 2021 at 0:30 comment added Ryan Budney @lemon314: the presentation will depend on the precise Heegaard splitting chosen. You get the vertical Heegaard splittings by tubing together neighbourhoods of singular fibers. So these presentations will be just a few Tietze moves away from the standard textbook presentations you get from thinking of them as Seifert fibered spaces.
Jul 3, 2021 at 0:07 comment added lemon314 Thank you very much for your answer. I wonder if there's a more explicit reference, as the paper doesn't spell out the presentation. If none shows up, I will accept this answer in couple of days. I'll look into Proposition 1.5, because although it's only about a very specific group, the method seems applicable in greater generality.
Jul 2, 2021 at 21:46 history answered Danny Ruberman CC BY-SA 4.0