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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)].
Remains
Seifert surgeries on M(R1, R2, R3)
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
2. Toroidal Seifert surgery
10 / 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 : Toroidal Seifert surgery
Recall: Each exceptional surgery is either:
• Reducible (conjectured:6 ∃ (Cabling Conjecture)),
• Toroidal,
• Seifert.
11 / 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 : Toroidal Seifert surgery
Recall: Each exceptional surgery is either:
• Reducible (conjectured:6 ∃ (Cabling Conjecture)),
• Toroidal,
• Seifert.
Remark [Eudave-Munoz (2002)]
They are not exclusive (i.e., there are non-empty intersection).
11 / 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 : Toroidal Seifert surgery
Recall: Each exceptional surgery is either:
• Reducible (conjectured:6 ∃ (Cabling Conjecture)),
• Toroidal,
• Seifert.
Remark [Eudave-Munoz (2002)]
They are not exclusive (i.e., there are non-empty intersection).
Theorem [Motegi (2003)]
A hyperbolic knot K with |Sym∗(K)| > 2has no toroidal Seifert surgery.
11 / 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 : Toroidal Seifert surgery
Recall: Each exceptional surgery is either:
• Reducible (conjectured:6 ∃ (Cabling Conjecture)),
• Toroidal,
• Seifert.
Remark [Eudave-Munoz (2002)]
They are not exclusive (i.e., there are non-empty intersection).
Theorem [Motegi (2003)]
A hyperbolic knot K with |Sym∗(K)| > 2has no toroidal Seifert surgery.
In particular, other than the trefoil knot,no two-bridge knots admit toroidal Seifert surgeries.
11 / 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
Results : Toroidal Seifert surgery
Theorem [I.-Jong (2010)]
Montesinos knots admit no toroidal Seifert surgeriesother than the trefoil knot.
Corollary
No hyperbolic Montesinos knot admit toroidal Seifert surgery.
12 / 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
Results : Toroidal Seifert surgery
Theorem [I.-Jong (2010)]
Montesinos knots admit no toroidal Seifert surgeriesother than the trefoil knot.
Corollary
No hyperbolic Montesinos knot admit toroidal Seifert surgery.
Remains
Atoroidal Seifert surgeries on M(R1, R2, R3)
12 / 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
Results : Toroidal Seifert surgery
Theorem [I.-Jong (2010)]
Montesinos knots admit no toroidal Seifert surgeriesother than the trefoil knot.
Corollary
No hyperbolic Montesinos knot admit toroidal Seifert surgery.
Remains
Atoroidal Seifert surgeries on M(R1, R2, R3)
Theorem [Wu (2009, 2010)]
If K = M(R1, R2, R3) admits an atoroidal Seifert surgery,then K = P (q1, q2, q3) or P (q1, q2, q3,−1)
with (|q1|, |q2|, |q3|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).
12 / 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
3. Cyclic/Finite surgery
13 / 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
cyclic / finite surgery
Problem
On (hyperbolic) knots in S3,determine all Dehn surgeries giving 3-manifoldswith cyclic or finite fundamental groups.
We call such surgeriescyclic surgeries / finite surgeries respectively.
14 / 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
cyclic / finite surgery
Problem
On (hyperbolic) knots in S3,determine all Dehn surgeries giving 3-manifoldswith cyclic or finite fundamental groups.
We call such surgeriescyclic surgeries / finite surgeries respectively.
Remark
· Such manifolds are all Seifert fibered.· On non-hyperbolic knots, such surgeries are classified.
14 / 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
Theorem
We give a complete classification ofcyclic / finite surgeries on Montesinos knots.
15 / 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
Theorem
We give a complete classification ofcyclic / finite surgeries on Montesinos knots.
Theorem [I.-Jong (2009)]
K : hyperbolic Montesinos knotK(r): the manifold obtained by surgery on K along r.
15 / 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
Theorem
We give a complete classification ofcyclic / finite surgeries on Montesinos knots.
Theorem [I.-Jong (2009)]
K : hyperbolic Montesinos knotK(r): the manifold obtained by surgery on K along r.
• If π1(K(r)) is cyclic,then K = P (−2, 3, 7) and r = 18 or 19.
15 / 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
Theorem
We give a complete classification ofcyclic / finite surgeries on Montesinos knots.
Theorem [I.-Jong (2009)]
K : hyperbolic Montesinos knotK(r): the manifold obtained by surgery on K along r.
• If π1(K(r)) is cyclic,then K = P (−2, 3, 7) and r = 18 or 19.
• If π1(K(r)) is acyclic finite,then K = P (−2, 3, 7) and r = 17, or
K = P (−2, 3, 9) and r = 22 or 23.
15 / 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
4. On alternating knots
16 / 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
Alternating knots
alternating knot
An alternating diagram = the crossings alternateunder, over, under, over, as you travel along the knot.
A knot is alternating if it admits an alternating diagram.
17 / 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
Alternating knots
alternating knot
An alternating diagram = the crossings alternateunder, over, under, over, as you travel along the knot.
A knot is alternating if it admits an alternating diagram.
Remark [Lickorish-Thistlethwaite]
A Montesinos knot is alternating if and only ifits reduced Montesinos diagram is alternating.
In particular, M(R1, . . . , Rl) is alternatingif R1, . . . , Rl have the same sign.
17 / 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
Results : atoroidal Seifert surgery
Theorem [I.-Jong-Mizushima]
If K = M(R1, R2, R3) with R1, R2, R3 > 0(i.e. K is alternating) admits an atoroidal Seifert surgery,
then K = P (a, b, c) with odd integers 3 ≤ a<b<c.
18 / 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
Results : atoroidal Seifert surgery
Theorem [I.-Jong-Mizushima]
If K = M(R1, R2, R3) with R1, R2, R3 > 0(i.e. K is alternating) admits an atoroidal Seifert surgery,
then K = P (a, b, c) with odd integers 3 ≤ a<b<c.
Theorem [Wu (2009, 2010)]
If K = M(R1, R2, R3) admits an atoroidal Seifert surgery,then K = P (q1, q2, q3) or P (q1, q2, q3,−1)
with (|q1|, |q2|, |q3|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).
18 / 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
Results : atoroidal Seifert surgery
Theorem [I.-Jong-Mizushima]
If K = M(R1, R2, R3) with R1, R2, R3 > 0(i.e. K is alternating) admits an atoroidal Seifert surgery,
then K = P (a, b, c) with odd integers 3 ≤ a<b<c.
Theorem [Wu (2009, 2010)]
If K = M(R1, R2, R3) admits an atoroidal Seifert surgery,then K = P (q1, q2, q3) or P (q1, q2, q3,−1)
with (|q1|, |q2|, |q3|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).
Corollary
An alternating hyperbolic Montesinos knot with length 3admits no Seifert surgery.
18 / 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
5. Remains
19 / 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
Remains
Atoroidal Seifert surgeries (not cyclic/finite) onP (q1, q2, q3) or P (q1, q2, q3,−1)
non-alternating and (|q1|, |q2|, |q3|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).
20 / 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
Remains
Atoroidal Seifert surgeries (not cyclic/finite) onP (q1, q2, q3) or P (q1, q2, q3,−1)
non-alternating and (|q1|, |q2|, |q3|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).
In progress : Exceptional surgeries on P (−2, n, n)
(joint work with I.D. Jong and Y. Kabaya)
20 / 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
Remains
Atoroidal Seifert surgeries (not cyclic/finite) onP (q1, q2, q3) or P (q1, q2, q3,−1)
non-alternating and (|q1|, |q2|, |q3|) = (2, ∗, ∗), (3, 3, ∗), (3, 4, 5).
In progress : Exceptional surgeries on P (−2, n, n)
(joint work with I.D. Jong and Y. Kabaya)
Thanks!
20 / 20