# Low Dimensional Topology

## February 16, 2013

### The Bridge Spectrum

Filed under: 3-manifolds,Heegaard splittings,Knot theory — Jesse Johnson @ 9:37 pm

A knot $K$ in a three-manifold $M$ is said to be in bridge position with respect to a Heegaard surface $\Sigma$ if the intersection of $K$ with each of the two handlebody components of the complement of $\Sigma$ is a collection of boundary parallel arcs, or if $K$ is contained in $\Sigma$. The bridge number of a knot $K$ in bridge position is the number of arcs in each intersection (or zero if if $K$ is contained in $\Sigma$) and the genus $g$ bridge number of $K$ is the minimum bridge number of $K$ over all bridge positions relative to genus $g$ Heegaard surfaces for $M$. The classical notion of bridge number is the genus-zero bridge number, i.e. bridge number with respect to a sphere in $S^3$, but a number of very interesting results in the last few years have examined the higher genus bridge numbers. Yo’av Rieck defined the bridge spectrum of a knot $K$ as the sequence $(b_0,b_1,b_2,\ldots)$ where $b_i$ is the genus $i$ bridge number of $K$ and asked the question: What sequences can appear as the bridge spectrum of a knot? (At least, I first heard this term from Yo’av at the AMS section meeting in Iowa City in 2011 – as far as I know, he was the first to formulate the question like this.)

So, first some preliminary facts: The bridge spectrum will always converge to $0$ after finitely many steps because a Heegaard surface for the complement $M \setminus N(K)$ will define a Heegaard surface $\Sigma$ for $M$ and $K$ can be isotoped into this $\Sigma$. The stabilizations of $\Sigma$ define Heegaard surfaces for $M$ that contain $K$ for every higher genus.

Second, we always have $b_{i+1} \leq b_i - 1$. This is because if $K$ is in bridge position with respect to $\Sigma$, we can attach a tube to $\Sigma$ along one of the bridge arcs of $K$ so that $K$ is still in bridge position with respect to the resulting surface $\Sigma'$. The number of bridge arcs thus decreases by one, but the genus of $\Sigma'$ is one greater than that of $\Sigma$. This construction is called a meridional stabilization. If the difference between $b_{i+1}$ and $b_i$ is greater than one then we say that the bridge spectrum has a gap.

So this limits the possible sequences, but still leaves a lot of room for how big the gaps between consecutive bridge numbers can be. For example, torus knots have arbitrarily high genus-zero bridge number but their genus-one bridge numbers are all zero. For higher genus $g$, Minsky-Moriah-Schleimer have constructed examples of knots with arbitrarily high genus $g$ bridge number, but genus-$(g+1)$ bridge number zero [1]. So these knots have very large gaps in their bridge spectra. One of the questions Yo’av asked at the AMS Special Session in 2011 was whether there are knots in $S^3$ whose bridge spectra have more than one gap of size greater than one.

Alex Zupan recently gave a positive answer to Yo’av’s question [2], and in fact proved something much stronger. Recall that a cabled knot is a knot $K_1$ that is contained in the boundary of a regular neighborhood of a second knot $K_0$. If $K_0$ is the unknot then any cable $K_1$ is a torus knot. We can then take a cable $K_2$ over $K_1$ and so on. If each $K_{i+1}$ is a cable over $K_i$, starting with the unknot $K_0$ then we will say that $K_n$ is an iterated torus knot. Alex shows that if you choose your cabled knot $K_n$ to be sufficiently complicated at each step in the construction then its bridge spectrum will have a gap at every step from $b_i$ to $b_n = 0$.

On the other hand, Alex also shows that if you take a genus 0, $b$-bridge knot with sufficiently high bridge distance then its bridge spectrum will have no gaps. In other words, the genus $i$ bridge number will be $b_i = b - i$ for each $i \leq b$. This is a really nice and very natural generalization of Maggy Tomova’s Theorem that a sufficiently high distance bridge surface always has minimal bridge number. There are only a few papers I know of that compare bridge surfaces of different genus, and this is by far the strongest result along these lines. It suggests a new perspective on bridge surfaces, in which surfaces of different genus fall very naturally into a single system, related by meridional stabilization and measured by the bridge spectrum. Alex’s paper generalizes much of the bridge surface/Heegaard surface machinery into this context and I expect that this will prove to be a very exciting new direction in the study of embedded surfaces.