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

Andrey Morozov

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.

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

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.
Paper Structure (8 sections, 44 equations, 4 figures)

This paper contains 8 sections, 44 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basics of $SU(N)$ calculus
  3. $SO(2n+1)$ calculus
  4. Fundamental representation of $SO(2n+1)$
  5. Symmetric representation of $SO(5)$
  6. Kauffmann polynomials in symmetric representation
  7. Conclusion
  8. Racah and $\mathcal{R}$-matrices for the symmetric representation of $SO(5)$

Figures (4)

  • Figure 1: Some representations of unknot with one, two or three strands and corresponding $\mathcal{R}$-matrix expressions.
  • Figure 2: Some representations of two unentangled unknots with two or three strands and corresponding $\mathcal{R}$-matrix expressions.
  • Figure 3: Some representations of the trefoil knot with two or three strands and corresponding $\mathcal{R}$-matrix expressions.
  • Figure 4: Figure-eight knot and its three-strand representation and corresponding $\mathcal{R}$-matrix expressions.