Bounding the stable genera of Heegaard splittings from below

Abstract

We describe for each positive integer k a 3‐manifold with Heegaard surfaces of genus 2k and 2k−1 such that any common stabilization of these two surfaces has genus at least 3k−1. We also find, for every k, a 3‐manifold with boundary admitting Heegaard splittings of genus k and k+1 whose stable genus is 2k, and for every positive n, a 3‐manifold that has n pairwise nonisotopic Heegaard splittings of the same genus all of which are stabilized.

Flipping bridge surfaces and bounds on the stable bridge number

“…In this paper we consider pairs of bridge splittings † and † 0 for .M; T / and study bridge splittings † 00 that can be obtained from both † and † 0 via stabilizations and perturbations. The results we obtain are similar but somewhat weaker than the results obtained by Johnson for Heegaard splittings in [5] due to the additional difficulties introduced by the presence of the knot.…”

“…Let be the projection map from f 1 OE˛0;ˇ0 to † 0 . By [5,Lemma 30], isotopy classes of loops in S project to isotopy classes in † 0 . Although we are now dealing with punctured surfaces the proof of this result is the same so we will not repeat it here.…”

“…Generalizing [5], for some fixed values of s and t we will say that † t is mostly above † 0 s if each component of † t \ H 0 s (if there are any) is contained in a disk or a once-punctured disk in † t . Similarly, we will say that † t is mostly below † 0 s if each component of…”

“…A long standing question in Heegaard splittings asks what is the minimal genus of † 00 in terms of the genera of † and † 0 . Examples of Heegaard splittings that required many stabilizations were presented by Bachman [1], Hass, Thompson and Thurston [3] and Johnson [5].…”

“…†; .H ; /; .H C ; C //. This value is called the flip Euler characteristic of † and it is analogous to the flip genus of a Heegaard splitting defined in [5]. We give a bound on this quantity in terms of the Euler characteristic of † and the distance of T with respect to † (Definition 4.1).…”