We explain how to construct knot and tangle invariants (such as Khovanov homology) by studying holomorphic vector bundles on certain compact, complex manifolds. Topologically these complex manifolds are just products of the same projective space $P^1$. Conjecturially, if one used Grassmannians $\mathrm{Gr}(k,n)$ instead of projective spaces this would give a series of new knot invariants analogous to Khovanov homology. \\ \noindent (Joint work with Joel Kamnitzer.)