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.

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