On Heegaard splittings of knot exteriors with tunnel number degenerations

Kanji Morimoto

paper

On Heegaard splittings of knot exteriors with tunnel number degenerations

arXiv:1310.7314

Abstract

Let $K_{1}, K_{2}$ be two knots with $t (K_{1}) + t (K_{2}) > 2$ and $t(K_1 # K_2)=2$. Then, in the present paper, we will show that any genus three Heegaard splittings of $E(K_1 # K_2)$ is strongly irreducible and that $E(K_1 # K_2)$ has at most four genus three Heegaard splittings up to homeomorphism. Moreover, we will give a complete classification of those four genus three Heegaard splittings and show unknotting tunnel systems of knots $K_1 # K_2$ corresponding to those Heegaard splittings.

13 pages, 12 figures