Tuesday, September 22, 2020
09/22/2020 - 4:00pm
Let S be a punctured surface. The $SL_2$-skein algebra of S is a non-commutative algebra, whose elements are represented by knots K in the thickened surface S x [0,1] subject to certain relations. The skein algebra is a quantum deformation of the $SL_2(C)$-character variety of S, depending on a deformation parameter q. Bonahon and Wong constructed an injective algebra map--called the quantum trace map--from the skein algebra of S into a simpler non-commutative algebra, the latter of which can be thought of as a quantum Teichmüller space of S. This mapping associates to a knot K in S x [0,1] a Laurent polynomial in q-commuting variables $X_i$, recovering in the classical case $q=1$ the polynomial expressing the traces of monodromies of hyperbolic structures on S when written in terms of Thurston's shear-bend coordinates for Teichmüller space. In the '00s, Fock and Goncharov, among others, developed a classical and quantum ''higher Teichmüller theory'', which naturally leads to a $SL_n$-version of this invariant.