On universal knot polynomials

Abstract

We present a universal knot polynomials for 2- and 3-strand torus knots in adjoint representation, by universalization of appropriate Rosso-Jones formula. According to universality, these polynomials coincide with adjoined colored HOMFLY and Kauffman polynomials at SL and SO/Sp lines on Vogel's plane, and give their exceptional group's counterparts on exceptional line. We demonstrate that [m,n]=[n,m] topological invariance, when applicable, take place on the entire Vogel's plane. We also suggest the universal form of invariant of figure eight knot in adjoint representation, and suggest existence of such universalization for any knot in adjoint and its descendant representation. Properties of universal polynomials and applications of these results are discussed.

26 pages, a number of misprints corrected, section 2.4 mainly sent to added Appendix B

References in corpus (13)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Colored HOMFLY polynomials of knots presented as double fat diagrams
• Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation
• Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations
• Colored knot polynomials for Pretzel knots and links of arbitrary genus
• Towards effective topological field theory for knots
• On colored HOMFLY polynomials for twist knots
• Colored knot polynomials. HOMFLY in representation [2,1]
• Colored HOMFLY polynomials that distinguish mutant knots
• On the defect and stability of differential expansion
• Reformulated invariants for non-torus knots and links
• Extending and quantising the Vogel plane
• Invariants of links of Conway type

Cited by in corpus (37)

• Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots
• Differential expansion and rectangular HOMFLY for the figure eight knot
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Split Casimir operator for simple Lie algebras, solutions of Yang-Baxter equations and Vogel parameters
• Differential expansion for link polynomials
• Split Casimir operator for simple Lie algebras in the cube of $\mathsf{ad}$-representation and Vogel parameters
• Tangle blocks in the theory of link invariants
• On Universal Quantum Dimensions
• Eigenvalue hypothesis for multi-strand braids
• Rectangular superpolynomials for the figure-eight knot
• Eigenvalue conjecture and colored Alexander polynomials
• Distinguishing Mutant Knots
• Checks of integrality properties in topological strings
• Knot polynomials for twist satellites
• Kerov functions for composite representations and Macdonald ideal
• On Hopf-induced deformation of topological locus
• Factorization of differential expansion for non-rectangular representations
• On exclusive Racah matrices $\bar{S}$ for rectangular representations
• A non-torus link from topological vertex
• Gaussian distribution of LMOV numbers
• Can tangle calculus be applicable to hyperpolynomials?
• On skew tau-functions in higher spin theory
• Irreducible representations of simple Lie algebras by differential operators
• $X_2$ series of universal quantum dimensions
• On Universal Eigenvalues of Casimir Operator
• The split Casimir operator and solutions of the Yang-Baxter equation for the $osp(M|N)$ and $s\ell (M|N)$ Lie superalgebras, higher Casimir operators, and the Vogel parameters
• Can Yang-Baxter imply Lie algebra?
• Defect and degree of the Alexander polynomial
• Hopf superpolynomial from topological vertices
• Khovanov-Rozansky cycle calculus for bipartite links
• Vogel's universality and the classification problem for Jacobi identities
• Torus knots in adjoint representation and Vogel's universality
• On universal quantum dimensions of certain two-parameter series of representations
• Projectors on invariant subspaces of representations ${\mathrm{ad}}^{\otimes 2}$ of Lie algebras $so(N)$ and $sp(2r)$ and Vogel parametrization
• Torus Knots in Adjoint Representation
• Split Casimir operator and universal formulation of the simple Lie algebras
• On ambiguity in knot polynomials for virtual knots