  Any knot $K$ in genus-$1$ $1$-bridge position can be moved by isotopy to lie
in a union of $n$ parallel tori tubed by $n-1$ tubes so that $K$ intersects
each tube in two spanning arcs, which we call a leveling of the position. The
minimal $n$ for which this is possible is an invariant of the position, called
the level number. In this work, we describe the leveling by the braid group on
two points in the torus, which yields a numerical invariant of the position,
called the $(1, 1)$-length. We show that the $(1, 1)$-length equals the level
number. We then find braid descriptions for $(1,1)$-positions of all $2$-bridge
knots providing upper bounds for their level numbers, and also show that the
$(-2, 3, 7)$-pretzel knot has level number two.
