Holomorphic families of knots

Abstract

Let $(M, [g])$ be a $3$-dimensional conformal manifold. The space of knots $\operatorname{Kn}(M)$ in $M$ is an infinite-dimensional manifold that is known to carry an almost complex structure. This structure is formally integrable by a result of Brylinski. We study finite dimensional holomorphic submanifolds in $\operatorname{Kn}(M)$. We define an holomorphic family of knots in $(M, [g])$ parametrised by a finite-dimensional complex manifold $(X, I_X)$, and construct several series of examples. We show that the base $(X, I_X)$ is K\"ahler, and if $X$ is compact, it is a projective variety of complex dimension at most $2$. In this case the conformal structure $[g]$ uniquely determines the complex structure $I_X$ and vice versa. We prove that if an holomorphic family of knots in $(M, [g])$ over a compact base $(X, I_X)$ defines a foliation on $\mathbf{S}(TM)$, then $(X, I_X) \simeq \mathbb{C}\mathbf{P}^1 \times \mathbb{C}\mathbf{P}^1$, the manifold $(M, [g])$ is conformally equivalent to either $S^3$ or $\mathbb{R}\mathbf{P}^3$ with round metric, and all knots are geodesic in some round metric in the class.