(joint work with Jun Murakami)
A famous open problem in knot theory is to relate quantum invariants such as the Jones polynomial to invariants from hyperbolic geometry. Both the Jones polynomial and hyperbolic geometry are closely connected to SL(2,C). More precisely, the Jones polynomial can be constructed from representations of the quantum group SLq(2) while PSL(2,C) is the isometry group of hyperbolic space. Our approach is to attempt to do classical geometry on the knot complement but replacing the isometry group by its quantum group equivalent SLq(2). Using non-commutative projective geometry we study representations of the fundamental group of the knot complement into the quantum group. This way a new braid group representation is found.