Algebraic & Geometric Topology

Stabilizing Heegaard splittings of high-distance knots

George Mossessian

Abstract

Suppose $K$ is a knot in $S3$ with bridge number $n$ and bridge distance greater than $2n$. We show that there are at most $2n n$ distinct minimal-genus Heegaard splittings of $S3 ∖ η(K)$. 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.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 6 (2016), 3419-3443.

Dates
Revised: 15 February 2016
Accepted: 21 April 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841265

Digital Object Identifier
doi:10.2140/agt.2016.16.3419

Mathematical Reviews number (MathSciNet)
MR3584263

Zentralblatt MATH identifier
06666105

Citation

Mossessian, George. Stabilizing Heegaard splittings of high-distance knots. Algebr. Geom. Topol. 16 (2016), no. 6, 3419--3443. doi:10.2140/agt.2016.16.3419. https://projecteuclid.org/euclid.agt/1510841265

References

• D Bachman, Heegaard splittings of sufficiently complicated $3$–manifolds, I: Stabilization, preprint (2009)
• J Hass, A Thompson, W Thurston, Stabilization of Heegaard splittings, Geom. Topol. 13 (2009) 2029–2050
• J Hempel, $3$–manifolds as viewed from the curve complex, Topology 40 (2001) 631–657
• J Johnson, Bounding the stable genera of Heegaard splittings from below, J. Topol. 3 (2010) 668–690
• J Johnson, Automorphisms of the three-torus preserving a genus-three Heegaard splitting, Pacific J. Math. 253 (2011) 75–94
• J Johnson, An upper bound on common stabilizations of Heegaard splittings, preprint (2011)
• J Johnson, M Tomova, Flipping bridge surfaces and bounds on the stable bridge number, Algebr. Geom. Topol. 11 (2011) 1987–2005
• T Kobayashi, Classification of unknotting tunnels for two bridge knots, from “Proceedings of the Kirbyfest” (J Hass, M Scharlemann, editors), Geom. Topol. Monogr. 2, Geom. Topol. Publ., Coventry (1999) 259–290
• J,N Mather, Stability of $C\sp{\infty }$ mappings, V: Transversality, Advances in Math. 4 (1970) 301–336
• K Morimoto, M Sakuma, On unknotting tunnels for knots, Math. Ann. 289 (1991) 143–167
• K Reidemeister, Zur dreidimensionalen Topologie, Abh. Math. Sem. Univ. Hamburg 9 (1933) 189–194
• H Rubinstein, M Scharlemann, Comparing Heegaard splittings of non-Haken $3$–manifolds, Topology 35 (1996) 1005–1026
• J Singer, Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc. 35 (1933) 88–111
• M Tomova, Multiple bridge surfaces restrict knot distance, Algebr. Geom. Topol. 7 (2007) 957–1006
• F Waldhausen, On irreducible $3$–manifolds which are sufficiently large, Ann. of Math. 87 (1968) 56–88