Can tangle calculus be applicable to hyperpolynomials?

Abstract

We make a new attempt at the recently suggested program to express knot polynomials through topological vertices, which can be considered as a possible approach to the tangle calculus: we discuss the Macdonald deformation of the relation between the convolution of two topological vertices and the HOMFLY-PT invariant of the 4-component link $L_{8n8}$, which both depend on four arbitrary representations. The key point is that both of these are related to the Hopf polynomials in composite representations, which are in turn expressed through composite Schur functions. The latter are further expressed through the skew Schur polynomials via the remarkable Koike formula. It is this decomposition that breaks under the Macdonald deformation and gets restored only in the (large $N$) limit of $A^{\pm 1}\longrightarrow 0$. Another problem is that the Hopf polynomials in the composite representations in the refined case are "chiral bilinears" of Macdonald polynomials, while convolutions of topological vertices involve "non-chiral combinations" with one of the Macdonald polynomials entering with permuted $t$ and $q$. There are also other mismatches between the Hopf polynomials in the composite representation and the topological 4-point function in the refined case, which we discuss.