Alternate Heegaard genus bounds distance

By M. Scharlemann, Maggy Tomova. 2005

127 citations recorded by the source corpus.

Suppose M is a compact orientable irreducible 3-manifold with Heegaard splitting surfaces P and Q. Then either Q is isotopic to a possibly stabilized or boundary-stabilized copy of P or the distance d(P)<= 2genus (Q). More generally, if P and Q are bicompressible but weakly incompressible connected closed separating surfaces in M then either P and Q can be well-separated or P and Q are isotopic or d(P)<= 2genus (Q).

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e8a392e435f99393445c07a74eef8d7021e92f53

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Tao Li
2. Juan Souto
3. Hossein Namazi
4. Y. Moriah
5. Yair N. Minsky
6. Saul Schleimer
7. Joseph Maher
8. Maggy Tomova

In Geometry and Topology

1. Joseph Maher
2. Juan Souto
3. Hossein Namazi
4. Saul Schleimer
5. Maggy Tomova
6. Tao Li
7. Y. Moriah
8. Yair N. Minsky

In Mathematical Physics

1. Juan Souto
2. Hossein Namazi
3. Saul Schleimer
4. Maggy Tomova
5. Tao Li
6. Joseph Maher
7. Y. Moriah
8. Yair N. Minsky

Citations in the corpus

• Multiple bridge surfaces restrict knot distance (2005, methodology): “The following definition was first used by Scharlemann and Tomova in [12].”
• High distance knots (2006, methodology): “Work of Scharlemann–Tomova [20] and Tomova [23; 22] links these two notions of complexity.”
• Random Heegaard splittings (2008, background): “By work of Hartshorn [5], Theorem 1.1 shows that the probability that M(wn) contains an incompressible surface of genus at most N tends to zero as n tends to infinity, and by Scharlemann and Tomova [21], the probability that the manifold has a Heegaard splitting of lower genus tends to zero as n ten”
• Heegaard Splittings and Pseudo-Anosov Maps (2009, methodology): “Theorem 1.8 is also due to Scharlemann–Tomova [ ScT ] but their methods are completely different from the ones used in the present paper.”
• Proximity in the curve complex: boundary reduction and bicompressible surfaces (2004, unknown): “Forthcoming work with M. Tomova ([ STo ]) will show how an augmented version of this theorem leads to the same conclusion as in Hartshorn’s theorem, not from an essential surface but from an alternate Heegaard surface.”
• Heegaard surfaces and the distance of amalgamation (2008, methodology): “5 basically follows from a theorem of Scharlemann and Tomova [26] and a theorem of Hartshorn [6], also see [12].”