Knot polynomials for twist satellites

Abstract

We begin the systematic study of knot polynomials for the twist satellites of a knot, when its strand is substituted by a 2-strand twist knot. This is a generalization of cabling (torus satellites), when the substitute of the strand was a torus knot. We describe a general decomposition of satellite's colored HOMFLY in those of the original knot, where contributing are adjoint and other representations from the "$E_8$-sector", what makes the story closely related to Vogel's universality. We also point out a problem with lifting the decomposition rule to the level of superpolynomials -- it looks like such rule, if any, should be different for positive and negative twistings.

8 pages

References in corpus (7)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Differential expansion and rectangular HOMFLY for the figure eight knot
• On rectangular HOMFLY for twist knots
• Differential expansion for link polynomials
• Rectangular superpolynomials for the figure-eight knot
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities
• HOMFLY for twist knots and exclusive Racah matrices in representation [333]

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• Planar decomposition of the HOMFLY polynomial for bipartite knots and links
• Khovanov polynomials for satellites and asymptotic adjoint polynomials
• Towards tangle calculus for Khovanov polynomials
• Planar decomposition of bipartite HOMFLY polynomials in symmetric representations
• Algebra of quantum $\mathcal{C}$-polynomials
• Evolution properties of the knot's defect
• Khovanov-Rozansky cycle calculus for bipartite links
• Khovanov--Rozansky matrix factorization reduction for bipartite links