EDITED: The figures below now give a surgery sequence starting with three curves and ending with one. 

[![surface with three curves and one surgery arc][1]][1]

[![surface with two curves and one surgery arc][2]][2]

[![surface with one curve][3]][3]

Here is an interesting question for you: in general, how long must the final curve be, as a function of the original integer vector?

  [1]: https://i.sstatic.net/rJp0U.png
  [2]: https://i.sstatic.net/F3Zpk.png
  [3]: https://i.sstatic.net/uBtSD.png