(Reposted from my old ldt blog)
Here’s the question: what are all the degree two, tunnel number one fibered knots? A knot in the 3-sphere is fibered if its complement is a surface bundle. An unknotting tunnel for a knot is an arc with its endpoints on the knot such that the complement of the knot and the arc is a genus two handlebody, and a knot has tunnel number one if it has an unknotting tunnel. Cho and McCullough [1] showed that every knot and unknotting tunnel is determined by a unique sequence of what they call cabling operations. (Finding the sequence of cablings is easy, showing that it’s unique is the hard part.) The degree of a knot and unknotting tunnel is the number of cabling operations necessary to produce the knot and tunnel.
The problem is that while the cabling move is very natural as a way of modifying the knot, it is unclear how to understand what it does to the homeomorphism type of the knot complement, in particular when it should produce a fibered complement. A sequence of cabling operations is described by a sequence of rational numbers and the long term motivation behind this question would be to list all sequences of a fixed length that describe fibered knots.
Degree one knots are precisely 2-bridge knots, and Gabai [2] classified the rational numbers that determine fibered two-bridge knots. The question above is really asking: When does cabling a 2-bridge knot and its upper or lower tunnel by a rational tangle produce a fibered knot? As a preliminary sub-question, one might ask if it is possible to get a fibered degree two knot by cabling a non-fibered 2-bridge knot. It is probably too optimistic to hope for a “no” answer to this second question.
[1] Follow link to the ArXiv.
[2] Genera of the arborescent links. Mem. Amer. Math. Soc. 59 (1986), no. 339, i–viii and 1–98.
