Holomorphic families of knots

Abstract

Let $(M, [g])$ be a $3$-dimensional conformal manifold. The space of knots $\mathrm{Kn}(M)$ 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 $\mathrm{Kn}(M)$. We give a definition of an holomorphic family of knots in $(M, [g])$ parametrised by a finite-dimensional complex manifold $(X, I_X)$, and construct several families of examples. We show that the base $(X, I_X)$ is Kähler, and if $X$ is compact, it is a projective variety of complex dimension at most $2$. Finally, we prove that if an holomorphic family of knots in $(M, [g])$ over a compact base $(X, I_X)$ defines a foliation on the spherisation of the tangent bundle of $M$, 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.