1-Bridge Braids in Solid Tori (Berge-Gabai knots), dual knots, & knot exteriors

Question

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!

Answer 1

For Question #1:

As you acknowledge, Berge already gives an answer to Question #1. Using his notation, this is summarized in Table 2 of his paper "The knots in $D^2 \times S^1$ which have nontrivial surgeries that yield $D^2 \times S^1$". Berge also already tells you how to translate from a G-pair to a standard braid form in his Lemma 2.2. So really, you just need to do the legwork to get it into a more common presentations of the dual.

Alternatively, a non-G-pair exhibition of the pairs of dual Berge knots is given in the setting of their tangle quotients in Baker-Buck, "The classification of rational subtangle replacements between rational tangles". Figures 3-6 show bandings (+isotopy) between tangles in plat form keeping track of the dual arc, starting from a tangle isotopic to the crossingless two cup tangle.

(In [Baker-Buck], since they’re dual, I had combined types III and V into one family, though I really shouldn’t have. With one exception, there’s no overlap of them (as noted in Table 2 of [Berge]). Referring back to my "Surgery descriptions and volumes of Berge knots II" one finds that in Figure 5 of [Baker-Buck]: Top row shows V to III, Bottom row shows III to V.)

For Question #2:

If the exteriors of $K_1$ and $K_2$ are both homeomorphic to the manifold $X$, then either

• (A) their meridians are on the same boundary component of $X$ in which case they are a pair of Berge-Gabai duals as in Question #1, or
• (B) their meridians are on different boundary components of $X$ in which case this has been partially addressed by Wu in "The classification of Dehn fillings on the outer torus of a 1-bridge braid exterior which produce solid tori".

In Theorem 3.6 of [Wu], Wu determines both the 1-bridge braids for which a filling of the outer torus boundary of the exterior produces a solid torus again and the slopes of these fillings. What’s still missing is the determination of the dual knot the resulting solid torus: Is it necessarily a 1-bridge braid? If so, which one?

Note that if you drop the assumption of 1-bridge braid, situation (B) becomes much more varied. You’re basically looking at two component links in lens spaces where each component is isotopic (individually) to a core of a Heegaard solid torus. For example in $S^3$ any two component link of unknots has this property.