A few months ago, I wrote a blog post about the interesting phenomenon that the tunnel number of a connect sum of two knots may be anywhere from one more than the sum of the tunnel numbers to a relatively small fraction of the sum of the tunnel numbers. Since then, a couple of related papers have been posted to the arXiv, so I thought that justifies another post on the subject. The first preprint I’ll discuss, by João Miguel Nogueira , gives new examples of knots in which the tunnel number degenerates by a large amount. The second paper, by Trent Schirmer  (who is currently a postdoc here at OSU), gives a new bound on the amount tunnel number and Heegaard genus can degenerate by under connect sum/torus gluing, respectively, in certain situations.
The degeneration ratio of a pair of knots , is , the complement of the ratio between the the tunnel number of the connect sum and the sum of the tunnel numbers of the original knots. In other words, this is the amount that the tunnel number drops from the expected amount, divided by the expected amount. Nogueira constructs knot pairs with tunnel numbers three and two, respectively, whose connect sum has tunnel-number three. Thus the degeneration ratio is , which is the highest value known for knots.
As I mentioned in the last post, Trent Schirmer found examples of pairs of two-component links such that the degeneration ratio under connect sums approaches . This is slightly higher than , but it turns out that getting the result for knots instead of links was a highly non-trivial problem. (Schirmer and Nogueira independently discovered roughly the same construction, but Trent wasn’t able to get it to work for knots.) Both of the examples use a structure called a free decomposition that was introduced by Tsuyoshi Kobayashi: Given a knot in a manifold , a free decomposition for is a collection of closed, embedded surface in , transverse to such that the complement in of is a collection of handlebodies. A bridge surface for is one example of a free decomposition, but in general free decompositions can be much more complicated. For example, it is not too difficult to construct a knot that has a free decomposition consisting of surfaces that are incompressible in the complement of the knot – You just have to arrange things so that the compressing disks for the handlebodies in the complement of must all run along arcs of .
The main difficulty in proving the degeneration ratio for Nogueira’s examples is showing that the original knots have tunnel number three and two, respectively. In general, there are not a lot of good methods for bounding tunnel number from below. (This is essentially the same problem as bounding Heegaard genus from below, which was one of the main obstacles to the rank vs. genus problem.) So, Nogueira’s paper includes a lot of very intricate work characterizing how Heegaard surfaces can intersect these particular free decompositions.
The second paper I mentioned above gives a new bound on tunnel number degeneration. Yo’av Rieck pointed out in the comments to the last post that there are already bounds in certain cases: Morimoto-Schultens have shown  that if and are small knots (i.e. there are no closed incompressible surfaces in their complements other than the boundary parallel torus) then the degeneration ratio is zero (the tunnel number of the connect sum is the sum of the original tunnel numbers.) Kobayashi-Rieck generalized this in , proving that the same bound holds if both knots are m-small, which means that for any incompressible surface in the complement of the knot, there is a compressing disk for the surface that intersects the knot in a single point (with merdional slope).
As shown by Nogueira’s examples (as well as by the first examples of tunnel number degeneration by Morimoto ), this bound can’t possibly hold in the case where one or both knots are not m-small. Thus Trent proves in  that if only one of , or is m-small then the tunnel number of the connect sum is greater than or equal to the maximum of the tunnel numbers of and . Translated into the language of degeneration ratio, this means that the degeneration ratio is at most . Trent’s proof again uses free decompositions, by noting that if the tunnel number drops under the connect sum then the resulting Heegaard splitting for the connect sum complement defines a free decomposition for each of the original knots. Trent uses these particular free decompositions to construct Heegaard splittings for the original knots whose genus is less than or equal to the Heegaard splitting of the connect sum complement. He also uses a clever trick to get a similar bound for gluing general three-manifolds along tori, but I’ll let you read his paper for the details of that one.
 Morimoto, Kanji; Schultens, Jennifer: Tunnel numbers of small knots do not go down under connected sum. Proc. Amer. Math. Soc. 128 (2000), no. 1, 269-278.
 Kobayashi, Tsuyoshi; Rieck, Yo’av: Heegaard genus of the connected sum of m-small knots. Comm. Anal. Geom. 14 (2006), no. 5, 1037-1077.
 K. Morimoto, There are knots whose tunnel numbers go down under connected sum, Proc. Amer. Math.
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