A knot in
${S}^{3}$
in genus-$1$$1$-bridge position
(called a
$\left(1,1\right)$-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
$\left(1,1\right)$-position.
After using them to verify previous calculations of the slope invariants for all tunnels of
$2$-bridge knots
and
$\left(1,1\right)$-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.

A knot in
${S}^{3}$
in genus-$1$$1$-bridge position
(called a
$\left(1,1\right)$-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
$\left(1,1\right)$-position.
After using them to verify previous calculations of the slope invariants for all tunnels of
$2$-bridge knots
and
$\left(1,1\right)$-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.

A knot in
${S}^{3}$
in genus-$1$$1$-bridge position
(called a
$\left(1,1\right)$-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
$\left(1,1\right)$-position.
After using them to verify previous calculations of the slope invariants for all tunnels of
$2$-bridge knots
and
$\left(1,1\right)$-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.

A knot in
${S}^{3}$
in genus-$1$$1$-bridge position
(called a
$\left(1,1\right)$-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
$\left(1,1\right)$-position.
After using them to verify previous calculations of the slope invariants for all tunnels of
$2$-bridge knots
and
$\left(1,1\right)$-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.

A knot in
${S}^{3}$
in genus-$1$$1$-bridge position
(called a
$\left(1,1\right)$-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
$\left(1,1\right)$-position.
After using them to verify previous calculations of the slope invariants for all tunnels of
$2$-bridge knots
and
$\left(1,1\right)$-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.

A knot in
${S}^{3}$
in genus-$1$$1$-bridge position
(called a
$\left(1,1\right)$-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
$\left(1,1\right)$-position.
After using them to verify previous calculations of the slope invariants for all tunnels of
$2$-bridge knots
and
$\left(1,1\right)$-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.