Torsion in skein modules and shared knot surgeries

Abstract

For every $g>0$, we construct a closed Haken 3-manifold $M_g$, whose only connected, incompressible surface is a non-separating surface of genus $g$ and discuss the torsion behavior of the Kauffman bracket skein module $\S (M_g,\mathbb{Z}[A^{\pm 1}])$. We also use properties of shared surgeries between knots to estimate (and often compute) the dimension of the Kauffman bracket skein module over $\Q(A)$, for families of hyperbolic 3-manifolds obtained by surgery on double twist knots. We have similar computations for some surgeries on several knot complements in the SnapPy census of cusped hyperbolic manifolds with triangulations up to 9 tetrahedra.