Abstract
We outline the current status of the differential expansion (DE) of colored knot polynomials i.e. of their Z–F decomposition into representation– and knot–dependent parts. Its existence is a theorem for HOMFLY-PT polynomials in symmetric and antisymmetric representations, but everything beyond is still hypothetical – and quite difficult to explore and interpret. However, DE remains one of the main sources of knowledge and calculational means in modern knot theory. We concentrate on the following subjects: applicability of DE to non-trivial knots, its modifications for knots with non-vanishing defects and DE for non-rectangular representations. An essential novelty is the analysis of a more-naive Z–FTw decomposition with the twist-knot F-factors and non-standard Z-factors and a discovery of still another triangular and universal transformation V, which converts Z to the standard Z-factors V−1Z=Z and allows to calculate F as F=VFTw.
Highlights
Knot polynomials are gauge invariant observables in 3d topological Chern-Simons (CS) theory [1], and they lie on the way from the well studied conformal blocks in 2d [2, 3] to the still-mysterious confinementcontrolling Wilson loops in 4d QCD [4]
There is no a priori reason for such a deformation to q = 1 to exist beyond single-line or single-row R, i.e. beyondsymmetric coloring — nothing to say about exact expression
Spectacular theory is already developed for twist knots
Summary
Knot polynomials are gauge invariant observables in 3d topological Chern-Simons (CS) theory [1], and they lie on the way from the well studied conformal blocks in 2d [2, 3] to the still-mysterious confinementcontrolling Wilson loops in 4d QCD [4]. The strange-looking conjecture of [14] is that supposed universality of (2) implies that MR is just the same for all other(!) knots This is a little less strange than seems, because for all rectangular representations R the product R ⊗ Rconsists only of diagonal composites, and is equivalent for the set of sub-diagrams. In this letter we provide an illustration and present coefficients F of a three-bridge knot This is no longer true for non-rectangular R, and the choice of MR in this case is still an open problem. We do not know HOMFLY-PT polynomials for sums of non-diagonal composite representations, which enter the “moduli space” MR in the case of non-rectangular representations R. That further efforts should be applied for the study of differential expansion (2), and we hope that, despite being tedious, it will attract attention that it deserves
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