Exceptional surgeries on Montesinos knots K.Ichihara Introduction Dehn surgery Exceptional surgery Montesinos knot Problem Known facts Toroidal Seifert surgery Known facts Result Cyclic/Finite surgery Cyclic/Finite surgery Results On alternating knots alternating knot Results Remains On exceptional surgeries on Montesinos knots Kazuhiro Ichihara Nihon University College of Humanities and Sciences joint works with In Dae Jong (OCAMI) Shigeru Mizushima (Tokyo Institute of Technology) V-JAMEX, Colima, Mexico, Sep 29, 2010 1 / 20
43
Embed
On exceptional surgeries on Montesinos knotsichihara/Research/... · Algebr. Geom. Topol. 9 (2009) 731{742. Preprint version, arXiv:0807.0905 K. Ichihara and I.D. Jong Toroidal Seifert
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
On exceptional surgerieson Montesinos knots
Kazuhiro Ichihara
Nihon UniversityCollege of Humanities and Sciences
joint works with
In Dae Jong(OCAMI)
Shigeru Mizushima(Tokyo Institute of Technology)
V-JAMEX, Colima, Mexico, Sep 29, 20101 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Preprints
This talk is based on
• K. Ichihara and I.D. JongCyclic and finite surgeries on Montesinos knotsAlgebr. Geom. Topol. 9 (2009) 731–742.Preprint version, arXiv:0807.0905
• K. Ichihara and I.D. JongToroidal Seifert fibered surgeries on Montesinos knots
Preprint, arXiv:1003.3517To apear in Comm. Anal. Geom.
• K. Ichihara, I.D. Jong and S. MizushimaSeifert fibered surgeries on alternating Montesinos knots
in preparation.
2 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
1. Introduction
3 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Dehn surgery on a knot
• K : a knot in the 3-sphere S3
• E(K): the exterior of K (:= S3−(open nbd. of K))
Dehn surgery on K
Gluing a solid torus V to E(K) to obtain a closed manifold.
4 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Dehn surgery on a knot
• K : a knot in the 3-sphere S3
• E(K): the exterior of K (:= S3−(open nbd. of K))
Dehn surgery on K
Gluing a solid torus V to E(K) to obtain a closed manifold.
K(r): the manifold obtained by Dehn surgery on K along r.
r ∈ Q ∪ {1/0}: surgery slope, corresponding to [ f(m) ] ,
where f : ∂V → ∂E(K), m: meridian of V .
4 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Exceptional surgery
Theorem [Thurston (1978)]
Dehn surgeries on a hyperbolic knot(i.e., knot with hyperbolic complement)
yielding a non-hyperbolic manifold are only finitely many.
5 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Exceptional surgery
Theorem [Thurston (1978)]
Dehn surgeries on a hyperbolic knot(i.e., knot with hyperbolic complement)
yielding a non-hyperbolic manifold are only finitely many.
Exceptional surgery
Dehn surgery on a hyperbolic knotyielding a non-hyperbolic manifold is called exceptional surgery.
5 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Exceptional surgery
Theorem [Thurston (1978)]
Dehn surgeries on a hyperbolic knot(i.e., knot with hyperbolic complement)
yielding a non-hyperbolic manifold are only finitely many.
Exceptional surgery
Dehn surgery on a hyperbolic knotyielding a non-hyperbolic manifold is called exceptional surgery.
An exceptional surgery is either:
• Reducible surgery (yielding a manifold. containing an essential S2)
• Toroidal surgery (yielding a manifold. containing an essential T2)
• Seifert surgery (yielding a Seifert fibered manifold.)
as a consequence of the Geometrization Conjectureestablished by Perelman (2002-03).
5 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Montesinos knot
Montesinos knot M(R1, . . . , Rl) in S3
A knot admitting a diagram obtained by putting rationaltangles R1, . . . , Rl together in a circle.
arcs on a 4-punctured sphere, and 12-tangle
length of the knot= minimal number of
rational tangles M(12 , 1
3 ,−23)
P (a1, · · · , an) = M( 1a1
, · · · , 1an
) : (a1, · · · , an)-pretzel knot.
6 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Problem
Problem
Classify all the exceptional surgerieson hyperbolic Montesinos knots.
7 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Problem
Problem
Classify all the exceptional surgerieson hyperbolic Montesinos knots.
Remark [Menasco], [Oertel], [Bonahon-Siebenmann]
Non-hyperbolic Montesinos knots are
T (2, n), P (−2, 3, 3)(=T (3, 4)), P (−2, 3, 5)(=T (3, 5)).
T (x, y) : the (x, y)-torus knot.
Remark
Dehn surgeries on torus knotshave been completely classified by Moser (1971).
7 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Known facts : Length other than 3
K : hyperbolic Montesinos knot with length l
8 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Known facts : Length other than 3
K : hyperbolic Montesinos knot with length l
• l ≤ 2 ⇒ K is a two-bridge knot.Exceptional surgeries for them are completely classified
[Brittenham-Wu (1995)].
8 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Known facts : Length other than 3
K : hyperbolic Montesinos knot with length l
• l ≤ 2 ⇒ K is a two-bridge knot.Exceptional surgeries for them are completely classified
[Brittenham-Wu (1995)].
• l ≥ 4 ⇒ K admits no exceptional surgery [Wu (1996)].
8 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Known facts : Length other than 3
K : hyperbolic Montesinos knot with length l
• l ≤ 2 ⇒ K is a two-bridge knot.Exceptional surgeries for them are completely classified
[Brittenham-Wu (1995)].
• l ≥ 4 ⇒ K admits no exceptional surgery [Wu (1996)].
Remains
Exceptional surgeries on M(R1, R2, R3) (i.e. l = 3)
8 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Known facts : Reducible / Toroidal surgery
• 6 ∃ reducible surgeries on Montesinos knots [Wu (1996)].
9 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Known facts : Reducible / Toroidal surgery
• 6 ∃ reducible surgeries on Montesinos knots [Wu (1996)].
• Toroidal surgeries on Montesinos knotsare completely classified [Wu (2006)].
9 / 20
Exceptionalsurgeries onMontesinos
knots
K.Ichihara
Introduction
Dehn surgery
Exceptionalsurgery
Montesinos knot
Problem
Known facts
ToroidalSeifert surgery
Known facts
Result
Cyclic/Finitesurgery
Cyclic/Finitesurgery
Results
On alternatingknots
alternating knot
Results
Remains
Known facts : Reducible / Toroidal surgery
• 6 ∃ reducible surgeries on Montesinos knots [Wu (1996)].
• Toroidal surgeries on Montesinos knotsare completely classified [Wu (2006)].