• DOI:10.1142/s0218216522500146
• Corpus ID: 231662286

On linear resolvability of universal quantum dimensions

@article{Avetisyan2022OnLR,
  title={On linear resolvability of universal quantum dimensions},
  author={M. Y. Avetisyan and R. L. Mkrtchyan},
  journal={Journal of Knot Theory and Its Ramifications},
  year={2022}
}

• M. Avetisyan, R. Mkrtchyan
• Published 21 January 2021
• Mathematics
• Journal of Knot Theory and Its Ramifications

In the study of finite (Vassiliev’s) knot invariants, Vogel introduced the so-called universal parameters, belonging to the projective plane, in particular parametrizing the simple Lie algebras by Vogel’s table. Subsequently, a number of quantities, such as some universal knot invariants and (quantum) dimensions of simple Lie algebras, have been represented in terms of these parameters, i.e., in the universal form. We prove that at the points from Vogel’s table all known universal quantum dimension formulae are linearly resolvable, i.e., yield finite answers even if these points are singular, provided one restricts them to the appropriate lines. We show that the same phenomenon takes place for another three distinguished points in Vogel’s plane — [Formula: see text] and [Formula: see text]. We also examine the same formulae on linear resolvability at the remaining 48 distinguished points in Vogel’s plane, which correspond to the so-called [Formula: see text]-objects. Among them, three points were found to be regular for all known quantum dimension formulae. Two of them happen to be sharing a remarkable similarity with the simple Lie algebras, namely, the universal formulae yield integer-valued outputs (dimensions) at the indicated points in the classical limit.