A knot in
in genus-
-bridge position
(called a
-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
-position.
After using them to verify previous calculations of the slope invariants for all tunnels of
-bridge knots
and
-tunnels
of torus knots, we obtain characterizations of the slope sequences of tunnels of
-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.