TITLE: Are torus knot groups linear?
QUESTION [8 upvotes]: The fundamental group $T(p,q)$ of the complement of a $(p,q)$-torus knot (in $S^3$) admits the presentation $\langle a, b \mid a^p=b^q \rangle $. Is $T(p,q)$ linear, i.e., is there a faithful homomorphism of $T(p,q)$ into a matrix group over some field?
For example, $T(2,3)$ and $T(3,3)$ are linear, since both can be embedded into the 3-strand braid group $B_3$. (Indeed, $T(2,3)=B_3$.)

REPLY [7 votes]: By the work of Agol, Liu, Przytycki-Wise and Wise we now know that fundamental groups of irreducible 3-manifolds which are not closed graph manifolds are `virtually special', in particular they are linear over $\Bbb{Z}$.
Oddly enough, the only open case for linearity are fundamental groups of certain closed graph manifolds.