Bounding the stable genera of Heegaard splittings from below

Abstract

We describe for each postive 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 show that for every positive $n$, there is a 3-manifold that has $n$ pairwise non-isotopic Heegaard splittings of the same genus all of which are stabilized.

24 pages, 3 figures

References in corpus (5)

• Flipping and stabilizing Heegaard splittings
• Stabilizations of Heegaard splittings of sufficiently complicated 3-manifolds (Preliminary Report)
• Automorphisms of the three-torus preserving a genus three Heegaard splitting
• Stable functions and common stabilizations of Heegaard splittings
• Heegaard Splittings of Twisted Torus Knots

Cited by in corpus (17)

• Flipping bridge surfaces and bounds on the stable bridge number
• On the classification of Heegaard splittings
• Heegaard splittings and open books
• Mapping class groups of Heegaard splittings
• Heegaard splittings of sufficiently complicated 3-manifolds I: Stabilization
• Calculating isotopy classes of Heegaard splittings
• Mapping class groups of once-stabilized Heegaard splittings
• One-sided and two-sided Heegaard splittings
• An upper bound on common stabilizations of Heegaard splittings
• A refinement of Johnson's bounding for the stable genera of Heegaard splittings
• Stabilizing Heegaard Splittings of High-Distance Knots
• Distance and the Goeritz groups of bridge decompositions
• Stabilization, amalgamation, and curves of intersection of Heegaard splittings
• Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds
• Finite presentations of the mapping class groups of once-stabilized Heegaard splittings
• Heegaard splittings and singularities of the product map of Morse functions
• On the union stabilization of two Heegaard splittings