S. Garoufalidis andT. T. Q. Lê [Geom. Topol. 9, 1253–1293 (2005;Zbl 1078.57012)] proved that the colored Jones function of a knot \(K\), i.e., the function which maps each positive integer \(n\) to the colored Jones polynomial of \(K\) associated with the \(n\)-dimensional irreducible representation of the quantum group \(U_q(sl_2)\), is \(q\)-holonomic in the sense that the function satisfies a linear \(q\)-difference equation with coefficients in \(\mathbb{Q}(q,q^n)\). The \(q\)-holonomicity of the colored Jones function implies that there is a minimal recurrence relation expressed by a noncommutative polynomial in two variables \(E\) and \(Q\) with coefficients in \(\mathbb{Z}[q,q^{-1}]\), where \(E\) corresponds to the shift of \(n\) by \(1\) and \(Q\) corresponds to multiplication of \(q^n\). This noncommutative polynomial is called the noncommutative \(A\)-polynomial.
S. Garoufalidis [Geometry and Topology Monographs 7, 291–309 (2004;Zbl 1080.57014)] conjectured that the noncommutative \(A\)-polynomial with \(q=1\) is equal to the \(A\)-polynomial introduced byD. Cooper, M. Culler, H. Gillet, D. D. Long andP. B. Shalen [Invent. Math. 118, No. 1, 47–84 (1994;Zbl 0842.57013)]. This conjecture is called the AJ-conjecture.
In the paper under review, by introducing a multi-certificate version of creative telescoping method, the authors explicitly compute the noncommutative \(A\)-polynomial for the twist knot with \(p\) full twist with \(-15\leq p\leq 15\). This result implies a new proof of the AJ-conjecture for those knots.

Reviewer: Kazuo Habiro (Kyoto)

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