Kazuhiro Ichihara - Boundary slopes nearly bound exceptional slopes

Abstract

For a hyperbolic knot in the 3-sphere, we say a non-meridional slope is exceptional if Dehn surgery on that slope results in a non-hyperbolic manifold. We provide evidence in support of two conjectures. The first (inspired by a question of Kimihiko Motegi) states that any exceptional surgery slope occurs in the interval bounded by the least and the greatest finite boundary slopes. Secondly, when there are exceptional surgeries, we conjecture there are (possibly equal) NIT (meaning that non-integral or toroidal) boundary slopes $b1\le b2$ so that the exceptional slopes lie in $[\lfloor b1\rfloor ;\lceil b2\rceil ]$. Moreover, if $\lceil b1\rceil \le \lfloor b2\rfloor$, the integers in the interval [⌈b1⌉; ⌊b2⌋] are all exceptional surgeries. This talk is based on a joint work with Thomas Mattman (California State University, Chico).

Contents

Exceptional surgery

We define a Dehn surgery. A surgery is called exceptional if

Any pair of cloased 3-mfd are connecte by a finite sequence of Dehn surgeries. (especially from the sphere) (Lickorish, Wallace, 1960’s)

Classification of 3mfd exist. They are either

• prime/JSJ decomposable
• Hyperbolic
• Seifert fiberd (Geometrization Theorem)

A knot is hyperbolic if its complement is hyperbolic.

Hyperbolic Dehn Surgery Theorem (Thurston) By using one Dehn surgeries we can achieve at most finitely many non-hyperbolic mfd.

Exeptional surgery: A dehn surgery on a hyperbolic knot is called exceptional if it yields a non-hyperbolic mfd. ⇒ For a given mfd, there are only finitely many exceptional surgeries.

Exceptional surgeries give reducible, toroidal or Seifert fibered 3-mfd.

Example: 8-knot If we do a surgery of the sphere along a 8-knot we can get the following:

• $0$-surgery toroidal
• $\pm 1,\pm 2,\pm 3$-surgey ⇒ Seifert fibered
• $\pm 4$-surgery toroidal

Including one which hasn’t been mentioned those give $10$ exceptional surgeries which give us $10$ exceptional surgery slopes (subset of rational numbers)

Essential Surface A surface $F$ is called essential if $F$ is is incompressible and $\partial$-incompressible.

Boundary slope

The boundary slope of an essential surface $F$ is defined as the slope of the boundary components of $F$.

Two conjectures

We gave some boundary slopes for a few famous knots. We notice that there are two boundary slopes $\in \mathbb{Q}$ such that all exceptional surfaces occur in between those.

This conjecture has been checked on many knots. A boundary slope is called NIT if it is non-integral or toroidal. Using this we make the previous conjecture stronger.

Presentation

• Definitions have been included on screen for multiple frames.