  A knot in the 3-sphere in genus-1 1-bridge position (called a (1,1)-position)
can be described by an element of the braid group of two points in the torus.
Our main results tell how to translate between a braid group element and the
sequence of slope invariants of the upper and lower tunnels of the
(1,1)-position. After using them to verify previous calculations of the slope
invariants for all tunnels of 2-bridge knots and (1,1)-tunnels of torus knots,
we obtain characterizations of the slope sequences of tunnels of 2-bridge
knots, and of a class of tunnels we call toroidal. The main results lead to a
general algorithm to calculate the slope invariants of the upper and lower
tunnels from a braid description. The algorithm has been implemented as
software, and we give some sample computations.
