While thinking of whether any web (spatial trivalent graph) without an embedded bridge can be realized as a branching locus of a finite branched cover over $S^3$, I realized that this problem is closely related to whether fundamental groups of web complements are residually finite or not. I heard that knot groups are residually finite. Are fundamental groups of web complements also residually finite?
That the fundamental groups of compact $3$-manifolds are residually finite was proved by Hempel in
Hempel, John, Residual finiteness for 3-manifolds. Combinatorial group theory and topology (Alta, Utah, 1984), 379–396, Ann. of Math. Stud., 111, Princeton Univ. Press, Princeton, NJ, 1987.
Hempel's proof required geometrization (which of course was not known at the time, but is now). As far as I know, there is no proof that does not depend on geometrization.