Bridge Spectrum

The topic of my thesis

Generalizing bridge number

Doll defined the genus g bridge number in 1991 and Zupan defined bridge spectrum in 2013. Below is my dissertation, which will appear in the Journal of Knot Theory and its Ramifications. To compute the genus g bridge number, we need to know how many times a knot has to intersect a genus g Heegaard surface. The pieces of the knot on either side of this surface must be trivial, that is to say, have no knotting. Below is an image of a blue genus one surface intersecting the Figure-8 knot, along with a green embedded disk, showing that the arc is trivial on the bottom of the surface. Notice that the green disk runs through the blue handle. The arc on top is also trivial, showing that the genus one bridge number of the Figure-8 knot is one.

See the paper

Bridge spectra of cables of 2-bridge knots