Knots in circle bundles are determined by their complements

We resolve a case of the oriented knot complement conjecture by showing that knots in an orientable circle bundle $N$ over a genus $g \geq 2$ surface $S$ are determined by their complements. We apply this to the setting of canonical knots in the unit and projective tangent bundles, which are knots that are the set of tangents to a closed curve on $S$. We show that canonical knots have homeomorphic complements if and only if their shadows differ by Reidemeister moves, (de)stabilizations, loops/cusps added by transvections, and mapping classes of $S$.

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