Holomorphic families of knots in 3-manifolds

The space of immersed oriented knots in a 3-manifold $M$ is an infinite-dimensional Fréchet manifold that has rich differential geometry. For instance, a choice of conformal structure on $M$ induces a complex structure operator on the space of knots: it acts on a normal vector field along a knot by rotation through the angle $\pi/2$ in positive direction. Brylinski showed that this operator is formally integrable. This makes it possible to consider families of knots parametrized by finite-dimensional complex manifolds that depend holomorphically on the parameter. I will discuss several examples of such families and prove a general structural result concerning the base of a holomorphic family: it is always a Kähler manifold, and if it is compact, then it is a projective variety of complex dimension at most two. Time permitting, I will also discuss the relation between holomorphic families of knots in the round sphere, the LeBrun twistor space, and superminimal surfaces in $\mathbb{H}^4$.

The talk is based on joint work with Rodion Déev (in progress).