Annuli in generalized Heegaard splittings and degeneration of tunnel number. (English) Zbl 0953.57002

Scharlemann, Martin; Schultens, Jennifer

Math. Ann. 317, No. 4, 783-820 (2000).

The main theorem of the paper shows how to cut a strongly irreducible generalized Heegaard splitting of a compact 3-manifold \(M\) along a family of essential annuli and create generalized Heegaard splittings for the resulting manifolds. This is applied to the behaviour of the tunnel number \(t\) under connected sums of knots. It is known that the difference between the sum of the tunnel numbers of the components \(K^1, \ldots, K^n\) of a composite knot and the tunnel number of the knot itself can be arbitrarily large. Thus one considers the degeneration ratio \(d(K^1, \ldots, K^n)\) which is defined as this difference divided by the sum of the tunnel numbers of the components, and the question arises if there is a uniform bound less than 1 on this degeneration ratio. Improving results of Kowng, it is shown that if all \(K^j\) are prime, then \(d(K^1,\ldots, K^n) \leq 2/3\), and that \(d(K^1,\ldots, K^n) \leq 3/5\) if none of the \(K^j\) is a 2-bridge knot.

Reviewer: B.Zimmermann (Trieste)

MSC:

57M25	Knots and links in the \(3\)-sphere (MSC2010)
57N10	Topology of general \(3\)-manifolds (MSC2010)

Keywords:

Heegaard splitting; degeneration ratio