High distance Heegaard splittings of 3-manifolds

Abstract

J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631-657] used the curve complex associated to the Heegaard surface of a splitting of a 3-manifold to study its complexity. He introduced the distance of a Heegaard splitting as the distance between two subsets of the curve complex associated to the handlebodies. Inspired by a construction of T. Kobayashi [T. Kobayashi, Casson-Gordon's rectangle condition of Heegaard diagrams and incompressible tori in 3-manifolds, Osaka J. Math. 25 (3) (1988) 553-573], J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631-657] proved the existence of arbitrarily high distance Heegaard splittings.In this work we explicitly define an infinite sequence of 3-manifolds {M n } via their representative Heegaard diagrams by iterating a 2-fold Dehn twist operator. Using purely combinatorial techniques we are able to prove that the distance of the Heegaard splitting of M n is at least n.Moreover, we show that π 1 (M n ) surjects onto π 1 (M n−1 ). Hence, if we assume that M 0 has nontrivial boundary then it follows that the first Betti number β 1 (M n ) > 0 for all n 1. Therefore, the sequence {M n } consists of Haken 3-manifolds for n 1 and hyperbolizable 3-manifolds for n 3.

High distance bridge surfaces

“…The construction presented in [3] to obtain the image of the Dehn twist operator also works in our setting. …”

Flipping bridge surfaces and bounds on the stable bridge number

“…†/. Such knots can be obtained for example by modifying the construction used in [2] to generate high distance Heegaard splittings by iterating a 2-fold Dehn twist operator.…”

Bridge distance and plat projections

“…Results on distances of Heegaard splittings in the case of closed surfaces or bridge surfaces for knot spaces typically split into two kinds. One presents either lower or upper bounds, for example, Evans [4], Blair-Tomova-Yoshizawa [2], Tao Li [7], Ichiahra-Saito [6] and Lustig-Moriah [8]. The other kind presents for given integers n a manifold/knot space with distance n, for example, Ido-Jang-Kobayashi [5] and Qiu-Zou-Guo [9].…”