We discuss work-in-progress constructing a quantum trace map for
the special linear group SL_n.  This is a kind of Reshetikhin-Turaev
invariant for knots in thickened punctured surfaces, coming from an
interaction between higher Teichmüller theory and quantum groups.

Let S be a punctured surface of finite genus.  The SL_2-skein algebra of S
is a non-commutative algebra whose elements are represented by framed links
K in the thickened surface S x [0,1] subject to certain relations.  The
skein algebra is a quantization of the SL_2(C)-character variety of S, where
the deformation depends on a complex parameter q.  Bonahon and Wong
constructed an injective algebra map, called the quantum trace, from the
skein algebra of S into a simpler non-commutative algebra which can be
thought of as a quantum Teichmüller space of S. This map associates to a
link K in S x [0,1] a Laurent q-polynomial in non-commuting variables X_i,
which in the specialization q=1 recovers the classical trace polynomial
expressing the trace of monodromies of hyperbolic structures on S when
written in Thurston's shear-bend coordinates for Teichmüller space.  In the
early 2000s, Fock and Goncharov, among others, developed a higher
Teichmüller theory, which should lead to a SL_n-version of this invariant.