Vector cross product structures on manifolds include symplectic, volume, G2- and Spin(7)-structures.
We show that the knot spaces of such manifolds have natural symplectic structures, and relate instantons
and branes in these manifolds to holomorphic disks and Lagrangian submanifolds in their knot spaces.
For the complex case, the holomorphic volume form on a Calabi–Yau manifold defines a complex vector
cross product structure. We show that its isotropic knot space admits a natural holomorphic symplectic
structure.We also relate the Calabi–Yau geometry of the manifold to the holomorphic symplectic geometry
of its isotropic knot space.