Torsion in Kauffman bracket skein module of a $4$-strand Montesinos knot exterior

Abstract

For an oriented $3$-manifold $M$, let $\mathcal{S}(M)$ denote its Kauffman bracket skein module over $\mathbb{Z}[q^{\pm\frac{1}{2}}]$. We show that $\mathcal{S}(M)$ admits torsion when $M$ is the exterior of the Montesinos knot $K(a_1/b_1,a_2/b_2,a_3/b_4,a_4/b_4)$ with each $b_i\ge 3$. This provides a negative answer to Problem 1.92 (G)-(i) in the Kirby's list, which asks whether $\mathcal{S}(M)$ is free when $M$ is irreducible and has no incompressible non-boundary parallel torus.