Link in $\mathbb{R}\mathbb{P}^3$ and the Topological Vertex

Abstract

We provide the first computations of colored unknots and Hopf link in $\mathbb{R}\mathbb{P}^3$ using both the topological vertex and its refinement. Our approach utilizes the toric Calabi-Yau threefold arising from the geometric transition of the cotangent bundle of $\mathbb{R}\mathbb{P}^3$ under the large $N$ duality. We find that the link invariants are series in the Kahler parameters of the toric Calabi-Yau manifold and the $q$-expansions of the rational functions of the series have positivity property. We conjecture that they are Poincare series of an infinite dimensional link homology theory for links in $\mathbb{R}\mathbb{P}^3$. We compare our results with that of the $S^3$ and speculate the consequences of the series nature of the invariants.