Racah matrices for the symmetric representation of the SO(5) group

TL;DR

This paper introduces Racah matrices for the symmetric representation of the SO(5) group, aiming to extend knot invariant calculations to the SO(N) case, which has been less explored compared to SU(N).

Contribution

It provides explicit R and Racah matrices for SO(5) and discusses how to compute the associated Kauffmann polynomials, advancing the understanding of SO(N) knot invariants.

Findings

Derived Racah matrices for the symmetric representation of SO(5).

Demonstrated how to compute Kauffmann polynomials using these matrices.

Highlighted challenges in extending Reshetikhin-Turaev approach to SO(2n+1).

Abstract

Approaches to calculate SU(N) colored knot invariants (HOMFLY-PT polynomials) are well and widely developed. However, SO(N) case is mostly forgotten. With this paper we want to start the discusion of how to generalize Reshetikhin-Turaev approach to the SO(2n+1) case and which difficutlies arise in this discussion. We provide R and Racah matrices for the symmetric representation of the SO(5) group and show how to find the corresponding Kauffmann polynomials.