Computing the symmetric $\mathfrak{gl}_1$-homology

The symmetric $\mathfrak{gl}_n$-homologies, introduced by Robert and Wagner, provide a categorification of the Reshetikhin--Turaev invariants corresponding to symmetric powers of the standard representation of quantum $\mathfrak{gl}_n$. Unlike in the exterior setting, these homologies are already non-trivial when $n=1$. Moreover, in this case, their construction can be greatly simplified. Our first aim is giving a down-to-earth description of the non-equivariant symmetric $\mathfrak{gl}_1$-homology, together with relations that hold in this setting. We then find a basis for the state spaces of graphs, and use it to construct an algorithm and a program computing the invariant for uncolored links.