Differential expansion and rectangular HOMFLY for the figure eight knot

Abstract

Differential expansion (DE) for a Wilson loop average in representation $R$ is built to respect degenerations of representations for small groups. At the same time it behaves nicely under some changes of the loop, e.g. of some knots in the case of $3d$ Chern-Simons theory. Especially simple is the relation between the DE for the trefoil $3_1$ and for the figure eight knot $4_1$. Since arbitrary colored HOMFLY for the trefoil are known from the Rosso-Jones formula, it is therefore enough to find their DE in order to make a conjecture for the figure eight. We fulfil this program for all rectangular representation $R=[r^s]$, i.e. make a plausible conjecture for the rectangularly colored HOMFLY of the figure eight knot, which generalizes the old result for totally symmetric and antisymmetric representations.

16 pages

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Cited by in corpus (23)

• Factorization of differential expansion for antiparallel double-braid knots
• On rectangular HOMFLY for twist knots
• Nimble evolution for pretzel Khovanov polynomials
• Differential expansion for link polynomials
• Eigenvalue hypothesis for multi-strand braids
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities
• Eigenvalue conjecture and colored Alexander polynomials
• Rectangular superpolynomials for the figure-eight knot
• Perspectives of differential expansion
• Distinguishing Mutant Knots
• Checks of integrality properties in topological strings
• Knot polynomials for twist satellites
• Extension of KNTZ trick to non-rectangular representations
• Orthogonal Polynomials in Mathematical Physics
• On exclusive Racah matrices $\bar{S}$ for rectangular representations
• On $6j$-symbols for symmetric representations of $U_q({\mathfrak{su}}_N)$
• HOMFLY for twist knots and exclusive Racah matrices in representation [333]
• KNTZ trick from arborescent calculus and the structure of differential expansion
• Pentad and triangular structures behind the Racah matrices
• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
• Algebra of quantum $\mathcal{C}$-polynomials
• Generalized hypergeometric series for Racah matrices in rectangular representations
• Evolution properties of the knot's defect