Torus knots in adjoint representation and Vogel's universality

Abstract

Vogel's universality gives a unified description of the adjoint sector of representation theory for simple Lie algebras in terms of three parameters $\alpha ,\beta ,\gamma$, which are homogeneous coordinates of Vogel's plane. It is associated with representation theory within the framework of Chern-Simons theory only, and gives rise to universal knot invariants. We extend the list of these latter further, and explain how to deal with the adjoint invariants for the torus knots $T[m,n]$ considering the case of $T[4,n]$ with odd $n$ in detail.

16 pages, LaTeX

References in corpus (25)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• On the Hecke algebras and the colored HOMFLY polynomial
• Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
• Evolution method and "differential hierarchy" of colored knot polynomials
• Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots
• On universal knot polynomials
• Chern-Simons Invariants of Torus Links
• Universality in Chern-Simons theory
• Nonperturbative universal Chern-Simons theory
• Casimir eigenvalues for universal Lie algebra
• Refined Chern-Simons versus Vogel universality
• Colored HOMFLY polynomial via skein theory
• Counting Higher Genus Curves with Crosscaps in Calabi-Yau Orientifolds
• On duality and negative dimensions in the theory of Lie groups and symmetric spaces
• Character Expansions for the Orthogonal and Symplectic Groups
• Split Casimir operator for simple Lie algebras, solutions of Yang-Baxter equations and Vogel parameters
• On Universal Quantum Dimensions
• Split Casimir operator for simple Lie algebras in the cube of $\mathsf{ad}$-representation and Vogel parameters
• Universal volume of groups and anomaly of Vogel's symmetry
• Can tangle calculus be applicable to hyperpolynomials?
• Partition function of Chern-Simons theory as renormalized q-dimension
• On Refined Vogel's universality
• Vogel's universality and Macdonald dimensions
• Hopf superpolynomial from topological vertices

Cited by in corpus (2)

• Vogel's universality and the classification problem for Jacobi identities
• Torus Knots in Adjoint Representation