HOMFLY for twist knots and exclusive Racah matrices in representation [333

A Morozov

doi:10.1016/j.physletb.2018.01.062

Abstract

Next step is reported in the program of Racah matrices extraction from the differential expansion of HOMFLY polynomials for twist knots: from the double-column rectangular representations R=[rr] to a triple-column and triple-hook R=[333]. The main new phenomenon is the deviation of the particular coefficient f[332][21] from the corresponding skew dimension, what opens a way to further generalizations.

Highlights

Calculation of Racah matrices is the long-standing, difficult and challenging problem in theoretical physics [1]. It is further obscured by the basis-dependence of the answer in the case of generic representations, but this ”multiplicity problem” is absent in the case of rectangular representations
Most important is the deviation from the coefficient f((2121|)11) from the skew dimension, even shifted – what is expressed by eq(23), see (31)
This new phenomenon explains the failure of previous naive attempts to write down an explicit general expression for F in arbitrary representation: an adequate substitute of the skew characters and appropriate generalization of the corresponding conjecture in [5] is needed for this

Summary

Introduction

Calculation of Racah matrices is the long-standing, difficult and challenging problem in theoretical physics [1]. Instead of (23) – as one more manifestation of discontinuity of the formulas, expressed in terms of hook variables This F -factor is the first, associated with the triple-hook diagram λ. To get an explicit formula we impose the polynomiality requirement on the correction factors η(μ22|11|00) to the naive analogue of (14)-(19) for 3-hook diagrams:. (note that a2 > 0 and b2 > 0 for 3-hook diagrams the shifts like N −→ N + (i1 + 1)δb2 − (j1 + 1)δa2) do not matter) and ξ = K (i1,j1|i2,j2|i3,j3) This is a Laurent polynomial at all m, satisfying (5).

Conclusion

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