The disk complex and 2-bridge knots

We give a purely combinatorial proof of a result of Kobayashi and Saeki that every genus one 1-bridge position of a nontrivial 2-bridge knot is a stabilization.

Connectivity of certain simplicial complexes associated to a handlebody

In this paper, we prove three simplicial complexes associate to a handlebody, which are separating disk complex, half disk complex and disk pants graph, are connected.

On the connectedness of subcomplexes of a disk complex

For a boundary-reducible 3-manifold M with ∂M a genus-g surface, we show that if M admits a genus-(g + 1) Heegaard surface S, then the disk complex of S is simply connected. Also we consider the connectedness of the complex of reducing spheres. We investigate the intersection of two reducing spheres for a genus-3 Heegaard splitting of ( torus ) × I.