A recent talk in our topology seminar by Trent Schirmer (who just joined OSU as a postdoc this year) got me thinking about three closely related (almost equivalent) problems in three-dimensional topology. Trent spoke about the following problem in knot theory: Given two knots in with tunnel numbers and , what would you expect the tunnel number of their connect sum to be? Recall that the tunnel number of a knot is the Heegaard genus of the knot complement minus one. With a little work, one can show that the tunnel number of the connect sum is at most . However, there are also examples where it is much lower and Trent has constructed links where the connect sum has tunnel number around . This is fairly interesting on its own, but it turns out there are (at least) two other situations with similar phenomenon that appear to have the same underlying reasons.
The second situation is as follows: Given two three-manifolds, each with a single torus boundary component, with Heegaard genera and , what is the Heegaard genus of the manifold that results from gluing the manifolds along the tori by a given homeomorphism? In this case, amalgamating the Heegaard splittings gives you a Heegaard splitting of genus , and for “most” gluing maps, this is the Heegaard genus of the resulting manifold . However, on the other end of the spectrum, Schultens-Weidmann have constructed examples where and for a specific gluing map, the Heegaard genus of the resulting manifold is . Note that the conjectured minimum answer for the tunnel number problem is , and I think a reasonable conjecture for the torus gluing problem would be , which is realized by the Schultens-Weidmann construction. (By the way, this is a beautifully simple construction but unfortunately I think describing it here would make this post too long (if it isn’t already.)) The best known lower bounds for both problems are much lower: roughly for connect sums by Scharlemann-Schultens  and for torus gluing, which follows from work of Ryan Derby-Talbot .
So, what’s the connection between these two problems? In the second we’re gluing two manifolds together along a pair of tori. In the first, we’re taking a connect sum, but if you think of a knot connect sum as a three-manifold operation, you’re really gluing along a pair of annuli in the boundaries of the knot complements. Tori and annuli behave very similarly, so it’s not surprising that we would find a similar phenomenon in these two settings. But then there’s the third setting that I mentioned.
A generalized Conway product of two knots/links is the following construction: Given knots or links and in separate copies of the three-sphere, choose an arc with endpoints in each knot (and interior disjoint from each knot). A regular neighborhood of each arc is a ball and each knot intersects this ball in two unknotted arcs. Remove each of these balls from the appropriate copy of and glue the complements together along the resulting sphere boundary components so that the four endpoints of each knot match up. (Note that there is a choice of a braid along the gluing sphere.) The result is a knot or link in , which is called a generalized Conway product.
Given knots with bridge numbers and , the bridge number of the connect sum will always be . However, the bridge number of a generalized Conway product can be much lower. In particular, Ryan Blair has constructed examples where the bridge number of the generalized Conway product of two knots with bridge numbers is . Ryan also proved that the bridge number of the generalized Conway product cannot be lower than , though is probably a reasonable conjecture. So again, we have very similar subadditivity.
But how is this related to gluing along a torus or annulus? The answer seems to be that when you compare bridge surfaces to Heegaard surfaces (which also covers tunnel number), you should work in the double branched cover. As I’ve written about before, a bridge surface lifts to a Heegaard surface in the double branched cover of branched over the knot. The genus of this Heegaard surface is the bridge number minus one. The four punctured sphere defined by the generalized Conway product lifts to a torus. So the subadditivity in this setting appears to be caused by the same type of phenomenon as in the other two, if you know how to look at it.
What I find really interesting is that each of the three settings suggests different techniques, and these different techniques have resulted in very different examples and very different lower bounds. But somehow, the underlying mechanics of the situation, which these different techniques try to exploit, seem to be very much the same.