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

TL;DR

This paper computes link invariants for unknots and Hopf links in real projective 3-space using topological vertex techniques, revealing series with positivity properties and proposing a new link homology conjecture.

Contribution

It provides the first computations of link invariants in $ ext{RP}^3$ via topological vertex and introduces a conjecture relating these invariants to an infinite-dimensional link homology theory.

Findings

Link invariants are series in Kahler parameters with positivity properties.

Results suggest a connection to an infinite-dimensional link homology theory.

Comparison with $S^3$ invariants offers new insights into link topology in $ ext{RP}^3$.

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.