Connected sums of knots and weakly reducible Heegaard splittings

Details Connected sums of knots and weakly reducible Heegaard splittings Connected sums of knots and weakly reducible Heegaard splittings

1999-12-21

arxiv.org/abs/math/9912171v3

Mathematics

Geometric Topology

26 pages 10 figures

Scientific paper

This paper studies the question of whether minimal genus Heegaard splittings of exterior spaces of knots which are connected sums are weakly reducible or not. Furthermore it is shown that the Heegaard splittings of the knots used by Morimoto to show that tunnel number can be sub-additive are all strongly irreducible. These are the first examples of strongly irreducible minimal genus Heegaard splittings of composite knots. We also give a characterization of when is a set of primitive annuli on a handlebody simultaneously primitive. This characterization is different from that given in [Go].

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