Let $K$ be a knot in a solid torus. Combining results of Berge and Gabai, we know that if $K$ admits a solid torus surgery, then $K$ is a 1-bridge braid. Using Gabai's result, we can figure out what the surgery slope is. If a 1-bridge braid admits a non-trivial solid torus surgery, the surgery slope is (almost) always unique (there is a unique 1-bridge braid which admits 2 non-trivial solid torus surgeries; most refer to it as K(7,2,4) in (w,b,t) notation).

If $B = K(7,2,4)$, we know that $B$ is equivalent to its dual, $B'$.

What is known about the relation between a 1-bridge braid and its dual in general? Namely:

**Question #1:** Given a 1-bridge braid $K$ that admits a solid torus surgery, when is it equivalent to its dual?

**Question #2**: Is it known when, given 1-bridge braids $K_1$ and $K_2$ such that $K_1 \not \simeq K_2$, their exteriors (in the solid torus $\mathbb{D}^2 \times S^1$) are homeomorphic?

I know Berge gives a necessary & sufficient condition for equivalence of knots, but I'm wondering if the answers to my above questions already exist in the literature in a more modern/non-G-pair-theoretic language.

Thanks!