Seifert surfaces for genus one hyperbolic knots in the 3-sphere (Q2279056)

The JSJ decomposition of a 3-manifold provides some insight into its fundamental structure. For instance, K. Motegi conjectured that there is a uniform upper bound on the number of components in the JSJ decomposition of the complements of hyperbolic knots in \(S^3\). This is a rather dramatic conjecture due to the unbounded complexity of various measures of knots (of particular note here is the knot genus), but the rigidity of hyperbolic 3-manifolds tends to restrict the structures within hyperbolic knot complements. The author cites a proof of concept: If we further restrict our attention to genus one knots, then a result of \textit{Y. Tsutsumi} [Interdiscip. Inf. Sci. 9, No. 1, 53--60 (2003; Zbl 1055.57011)] states that surgery along the standard longitude of a genus one hyperbolic knot produces a 3-manifold whose JSJ decomposition is comprised of at most seven components. Tsutsumi's result favors Motegi's conjecture; but even so, this is a result regarding a very specific class of surgeries along a very specific class of knots in \(S^3\). The author generalizes this result substantially to the complements of arbitrary genus one hyperbolic knots in \(S^3\), and he succeeds in further refining the result to an upper bound of five components. This is made sharp by exhibiting an infinite family of knots that realize this five-component bound (and another similarly constructed family realizing the four-component case). This produces yet another, stronger result in favor of Motegi's conjecture. The author uses many standard techniques familiar to a broad audience, but he presents them in a delightfully intuitive yet careful manner; and this dramatically improves the pacing and readability of the lengthy article. The results follow a natural progression from geometrically establishing an upper bound of six components and then through algebraically ruling out the six-component case. These detailed arguments require detailed illustrations to explain them, and these have a tendency to be rather cluttered and uninformative; but the author does an excellent job of providing the necessary details without the clutter. Overall, this is a rather impressive consolidation and clever application of many classical combinatorial, algebraic, and geometric results and techniques. Since it appears the author's primary direction for where he wants to take this work is to extend it to a more general result, I will remark that the techniques in the current article are likely insufficient to be able to do so. A more general result still seems a bit far off. For hyperbolic knot complements, their JSJ decompositions are defined via Seifert surfaces; and this amounts to filling the knot complement with a maximal collection of mutually disjoint Seifert surfaces of the knot. Since the Seifert surfaces of genus one knots are punctured tori, some components of the JSJ decompositions involved (or gluings thereof) must be thick, punctured tori; that is, genus-two handlebodies. Herein lies the major difficulty with being able to extend this result: Results for genus-two surfaces and handlebodies tend to be true only for this genus. Many arguments in this article require the use of properties that occur solely for genus-two surfaces and handlebodies. Therefore, if this result is to be extended beyond the genus-one sandbox, I believe it would be likely to require the implementation of less genus-dependent lines of argument to establish the initial upper bound. Regardless of my views about the possibility of extending these results and methods, the author has delivered an excellent article that deserves some attention. Perhaps my assessment is short-sighted and these results and techniques might extend in some clever way to the higher-genus regime, but this is certainly a structure result worth following.