#### Volume 16, issue 6 (2016)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Stabilizing Heegaard splittings of high-distance knots

### George Mossessian

Algebraic & Geometric Topology 16 (2016) 3419–3443
##### Abstract

Suppose $K$ is a knot in ${S}^{3}$ with bridge number $n$ and bridge distance greater than $2n$. We show that there are at most $\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)$ distinct minimal-genus Heegaard splittings of ${S}^{3}\setminus \eta \left(K\right)$. These splittings can be divided into two families. Two splittings from the same family become equivalent after at most one stabilization. If $K$ has bridge distance at least $4n$, then two splittings from different families become equivalent only after $n-1$ stabilizations. Furthermore, we construct representatives of the isotopy classes of the minimal tunnel systems for $K$ corresponding to these Heegaard surfaces.

##### Keywords
knot complement, high distance, common stabilization, Heegaard splitting, tunnel system
Primary: 57M25
Secondary: 57M27