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Stavros Garoufalidis and Christoph Koutschan

Abstract

Our goal is to compute the minimal-order recurrence of the colored Jones polynomial of the$7_4$ knot, as well as for the first four double twist knots. As a corollary, we verify the AJ Conjecture for the simplest knot$7_4$ with reducible nonabelian$SL\left(2,ℂ\right)$ character variety. To achieve our goal, we use symbolic summation techniques of Zeilberger’s holonomic systems approach and an irreducibility criterion for$q$–difference operators. For the latter we use an improved version of the qHyper algorithm of Abramov–Paule–Petkovšek to show that a given$q$–difference operator has no linear right factors. En route, we introduce exterior power Adams operations on the ring of bivariate polynomials and on the corresponding affine curves.

Keywords

Mathematical Subject Classification 2010

References

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