Uniqueness of minimal bridge surface

Question

Let $K \subset S^3$ be a knot and $P,Q \subset S^3$ be bridge spheres for $K$ with minimal bridge numbers (i.e. $|P \cap K| = |Q \cap K| = 2 b(K)$ where $b(K)$ is the bridge number of $K$). Does it follow that there is an isotopy of $S^3$ that preserves $K$ but takes $P$ to $Q$?

Answer 1

I believe the answer is no. The introduction to the following paper is relevant:

• Jang, Yeonhee; Kobayashi, Tsuyoshi; Ozawa, Makoto; Takao, Kazuto. A knot with destabilized bridge spheres of arbitrarily high bridge number. J. Lond. Math. Soc. (2) 93 (2016), no. 2, 379–396. arXiv:1501.06263.

All bridge spheres are equivalent up to stabilization. In the introduction of the above paper, they survey results on when destabilized bridge spheres are equivalent up to destabilization. In particular, the following examples (names are links to papers) have non-unique destabilized bridge spheres:

1. Birman gave a composite knot with two non-isotopic destabilized 3-bridge spheres.
2. Montesinos gave a prime knot with two non-isotopic destabilized 3-bridge spheres.
3. Johnson and Tomova gave a knot with two non-isotopic destabilized bridge spheres which are far apart in the sense of stable equivalence.
4. Jang gave a knot with four destabilized 3-bridge spheres which are pairwise non-isotopic.

The first linked paper also lists a number of cases where bridge spheres are unique, such as the unknot, rational knots, torus knots, and cables of meridionally small knots with unique bridge spheres.