Eigenvalue conjecture and colored Alexander polynomials

Abstract

We connect two important conjectures in the theory of knot polynomials. The first one is the property Al_R(q) = Al_{[1]}(q^{|R|}) for all single hook Young diagrams R, which is known to hold for all knots. The second conjecture claims that all the mixing matrices U_{i} in the relation {\cal R}_i = U_i{\cal R}_1U_i^{-1} between the i-th and the first generators {\cal R}_i of the braid group are universally expressible through the eigenvalues of {\cal R}_1. Since the above property of Alexander polynomials is very well tested, this relation provides a new support to the eigenvalue conjecture, especially for i>2, when its direct check by evaluation of the Racah matrices and their convolutions is technically difficult.

5 pages

References in corpus (7)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• 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
• Racah matrices and hidden integrability in evolution of knots
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• Factorization of differential expansion for antiparallel double-braid knots

Cited by in corpus (13)

• Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations
• Eigenvalue hypothesis for multi-strand braids
• Multiplicity-free $U_q(s{l}_N)$ 6-j symbols: relations, asymptotics, symmetries
• Distinguishing Mutant Knots
• Colored Alexander polynomials and KP hierarchy
• A new symmetry of the colored Alexander polynomial
• Harer-Zagier formulas for knot matrix models
• Interplay between symmetries of quantum 6-j symbols and the eigenvalue hypothesis
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• A novel symmetry of colored HOMFLY polynomials coming from $\mathfrak{sl}(N|M)$ superalgebras
• Direct proof of one-hook scaling property for Alexander polynomial from Reshetikhin-Turaev formalism
• Tug-the-hook symmetry for quantum 6j-symbols
• Multistrand Eigenvalue conjecture and Racah symmetries