Geometry and Topology Seminar

In this talk we will be interested in knots in orientable circle bundles N over a genus g > 1 surfaces. Examples of such knots arise naturally when N is the unit tangent bundle of a surface and the knot is the set of tangents to a curve on the surface. We call such knots canonical or Legendrian. In joint work with Tommaso Cremaschi, we show that knots in N are determined by their complements, a special case of the oriented knot complement conjecture. In the setting of unit tangent bundles, this demonstrates 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 the surface.