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

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57N10 Topology of general $$3$$-manifolds (MSC2010)
##### Keywords:
Heegaard splitting; degeneration ratio
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