All fibered links can be constructed from the unknot by a sequence of operations called plumbing (and then de-plumbing) Hopf bands. Interestingly, if a fibered link has an unknotting tunnel that happens to lie in the fiber, then plumbing a Hopf band along the tunnel results in a new fibered link that is again tunnel number one. Natural questions are whether this restricted plumbing can always be performed, and whether this is sufficient to construct all tunnel one, fibered links. I will answer the first question affirmatively, and discuss progress towards answering the second.

For more information, contact Khalid Bou-Rabee khalidb[at]umich(dot)edu