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Jan 25, 2018 at 20:03 comment added Chris Gerig It is an obstruction-theoretic calculation, that I have not unraveled but I refer to the explanation in Turaev's paper "Euler structures, nonsingular vector fields, and torsions of Reidemeister type", page 13 (it's a more general setup: your scenario is $m=3$ and uses the Heegaard decomposition).
Jan 25, 2018 at 6:45 comment added Tom Church I don't know anything about this field, so I can't answer your question. But I appreciate the work you put into writing it in such a clear and understandable way, so I hope someone else is able to.
Jan 24, 2018 at 22:21 history asked cjackal CC BY-SA 3.0