[Paper Review] Heegaard splittings and virtually Haken Dehn filling

Joseph D. Masters, William W. Menasco+1 more|ArXiv.org|Oct 27, 2002

Geometric and Algebraic Topology26 references3 citations

TL;DR

This paper demonstrates that the 3-fold cyclic cover of the exterior of the twist knot $K_{2n+1}$ is large, containing an essential genus-two surface. Using Heegaard splittings and compression techniques, the authors show that Dehn fillings of slope $3p/q$ with $|p| > 1$ yield virtually Haken 3-manifolds, providing explicit families of non-Haken but virtually Haken manifolds, supporting the virtual Haken conjecture.

ABSTRACT

We use Heegaard splittings to give some examples of virtually Haken 3-manifolds.

Motivation & Objective

• To construct explicit examples of non-Haken 3-manifolds that are virtually Haken, supporting the virtual Haken conjecture.
• To demonstrate that certain Dehn fillings of twist knot exteriors yield virtually Haken manifolds through covering space techniques.
• To show that the 3-fold cyclic cover of the exterior of $K_{2n+1}$ is large, containing essential genus-two surfaces.
• To apply Heegaard splitting and compression techniques to lift and stabilize surfaces in covering spaces.
• To extend results from Casson and Gordon on lifted surfaces to new families of 3-manifolds.

Proposed method

• Construct a Heegaard splitting of the knot exterior $M_n$ using a genus-two handlebody and a compression body.
• Lift the Heegaard splitting to the 3-fold cyclic cover $\tilde{M}_n$, preserving the surface structure and disk systems.
• Use the multi-handle addition theorem from [L] to verify incompressibility of surfaces formed by compressing the lifted Heegaard surface.
• Analyze the Whitehead graph of disk systems to verify connectivity and absence of cut vertices, ensuring incompressibility.
• Lift the longitude $\lambda$ of $M_n$ to $\tilde{M}_n$ and show it can be isotoped into the essential surface $S$.
• Apply the covering space correspondence: Dehn fillings of slope $3p/q$ on $M_n$ lift to fillings of slope $p/q$ on $\tilde{M}_n$, preserving essentiality.

Experimental results

Research questions

• RQ1 Can Heegaard splittings be used to construct essential surfaces in finite covers of non-Haken 3-manifolds?
• RQ2 Do Dehn fillings of slope $3p/q$ on twist knot exteriors yield virtually Haken 3-manifolds?
• RQ3 Is the 3-fold cyclic cover of $M_n$, the exterior of $K_{2n+1}$, large for all $n > 0$?
• RQ4 Can lifted surfaces from a knot exterior be compressed to yield essential surfaces in the cover?
• RQ5 What conditions on Dehn filling slopes ensure that the resulting manifold is virtually Haken?

Key findings

• The 3-fold cyclic cover of the exterior of $K_{2n+1}$ is large for every $n > 0$, containing an essential genus-two surface.
• Dehn fillings of $M_n$ with slope $3p/q$, where $(3p, q) = 1$ and $|p| > 1$, yield virtually Haken 3-manifolds.
• The essential genus-two surface in the cover remains incompressible under all Dehn fillings of slope $p/q$ with $|p| > 1$.
• The longitude $\lambda$ of $M_n$ can be isotoped into the essential surface in the 3-fold cover, confirming its geometric significance.
• The method generalizes to all $n > 0$, with the proof for $n=1$ serving as a template for the general case.
• The use of Whitehead graphs and the multi-handle addition theorem confirms incompressibility of the compressed surface in the cover.