Dehn surgery on knots：survey

Kimihiko Motegi (Nihon University)

Diﬀerential Topology 22
19 March 2022

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

目標

デーン手術の地図を描く！

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

1904：Poincaré described a remarkable
3–manifold which has a trivial homology,
but whose fundamental group is nontrivial.
This 3–manifold is now called
the Poincaré homology 3–sphere, and it clariﬁes thediﬀerence between “homology” and “homotopy”.

Poincaré conjecture
A closed 3–manifold with trivial fundamental group is homeomorphic to S 3

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

1907：Dehn tried to construct a homology
3–sphere which is not S 3 by identifying
boundaries of two knot exteriors suitably.
However, he was not able to show theresultant 3–manifold is not S 3 .
1910：Dehn introduced a new idea what wecall “Dehn surgery”.

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Dehn surgery
K: a knot in the 3-sphere S 3
E(K)

λ
μ
γ
removeglue

γ = pμ + qλ

p
q

N(K)

μ

K( pq ): a 3-manifold obtained from S 3 by pq -Dehn surgery on K

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Dehn surgery
K: a knot in the 3-sphere S 3
E(K)

λ
μ
γ
removeglue

γ = pμ + qλ

p
q

N(K)

μ

K( pq ): a 3-manifold obtained from S 3 by pq -Dehn surgery on K
The result of Dehn surgery is uniquely determined by the image of the meridianof the glued solid torus (independent of the image of the longitude)!

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Dehn surgery
K: a knot in the 3-sphere S 3
E(K)

λ
μ
γ
removeglue

γ = pμ + qλ

p
q

N(K)

μ

K( pq ): a 3-manifold obtained from S 3 by pq -Dehn surgery on K
The result of Dehn surgery is uniquely determined by the image of the meridianof the glued solid torus (independent of the image of the longitude)!

We call pq a surgery slope.
The gluing solid torus is often called pq -Dehn ﬁlling.
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Dehn surgery
K: a knot in the 3-sphere S 3
E(K)

λ
μ
γ
removeglue

γ = pμ + qλ

p
q

N(K)

μ

K( pq ): a 3-manifold obtained from S 3 by pq -Dehn surgery on K
The result of Dehn surgery is uniquely determined by the image of the meridianof the glued solid torus (independent of the image of the longitude)!

We call pq a surgery slope.
The gluing solid torus is often called pq -Dehn ﬁlling.
In the following, we consider nontrivial surgery pq ̸= 01
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

A little bit about group theory

G(K) = π 1(E(K))π 1 (K(p/q)) = G(K)
G(K)

< <
μ

< <
μpλq

~ π (K(1/0)) = π (S3 ) = {1}
=
1
1

< <

G(K) = μφweight 1<<

H1(E(K)) = Z = μKimihiko Motegi (Nihon University)< <π 1 (K(p/q)) = φ( μ )Dehn surgery on knots：survey

H 1 (K(p/q)) = Z

pZ

Diﬀerential Topology 22 19 March 2022

Examples

Lens space
Let K be the trivial knot O. Then O(p/q) is a lens space L(p, q).
In particular, O(1/n) is the 3–sphere for any integer n.

Homology 3–spheres
For any knot K, K(1/n) is a homology 3–sphere for any integer n.
If K is a trefoil knot T3,2 , then T3,2 (1) is the Poincaré homology 3–sphere.

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

One may easily generalize Dehn surgery on “knots” to Dehn surgery on“links”.
E(L)removeglue

glue

removeTheorem (Lickorish 1962, Wallace 1960)
Any orientable closed 3–manifold can be obtained from S 3 by Dehn
surgery on a “link”.

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

integral Dehn surgery
=

links

framed surgery

closed oriented
3-manifold

surjective!
(Lickorish-Wallace)

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Lickorish-Wallace + Kirby
framed links

Dehn surgery

closed oriented
3-manifold

surjective!
(Lickorish-Wallace
1960-1962)

framed links
Kirby-move(Kirby 1978)

Kimihiko Motegi (Nihon University)

Dehn surgery

closed oriented
3-manifold

not injective

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Lickorish-Wallace + Kirby-Fenn-Rourke-Rolfsen
links withrational slopes

Dehn surgery

closed oriented
3-manifold

surjective!
(Lickorish-Wallace
1960-1962)

links withrational slopes
Kirby-Rolfsen-move

Dehn surgery

closed oriented
3-manifold

not injective(Fenn-Rourke 1979)(Rolfsen 1984)

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Lickorish-Wallace + Kirby-Fenn-Rourke-Rolfsen
links withrational slopes

Dehn surgery

closed oriented
3-manifold

surjective!
(Lickorish-Wallace
1960-1962)

links withrational slopes
Kirby-Rolfsen-move

Dehn surgery

closed oriented
3-manifold

not injective(Fenn-Rourke 1979)(Rolfsen 1984)

We will focus on Dehn surgery on knots in S 3 .
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Problems in study of Dehn surgery

Given a knot K ⊂ S 3 and r ∈ Q, we obtain a closed oriented 3–manifold
K(r).
knots

Dehn surgery

closed oriented
3-manifold

More explicitly we may regard Dehn surgery as a map:
D : {knots in S 3 } × Q → {oriented closed 3–manifolds}
D(K, r) = K(r)Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Exceptional surgery problem
D : {knots in S 3 } × Q → {oriented closed 3–manifolds}
D(K, r) = K(r)Consider a classC∗ ⊂ {oriented closed 3–manifolds}
with “property ∗”.
Describe K and/or r such that D(K, r) ∈ C∗ .
In general, we take a “property ∗” as an exceptional one which describe akind of “degeneration” happens, so we call a problem of this type an
exceptional surgery problem.
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Surjectivity (realization) problem
D : {knots in S 3 } × Q → {oriented closed 3–manifolds}
D(K, r) = K(r)Recall ﬁrst that any 3–manifold obtained by a Dehn surgery on a knot in
S 3 has the fundamental group with weight one.
So we focus on
W1 = {closed, orientable 3–manifolds with weight one fundamental group}
and ask
Is D : {knots in S 3 } × Q → W1 surjective?
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Injectivity problem

D : {knots in S 3 } × Q → {oriented closed 3–manifolds}
D(K, r) = K(r)We may ask:
Is the map D : {knots in S 3 } × Q → {oriented closed 3–manifolds}
injective.
As we will discuss later, the injectivity problem may be divided into two
more speciﬁc problems according as we ﬁx a knot K or a surgery slope r.

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Property P conjecture
In the exceptional surgery problem, let us considerC∗ = {simply connected, closed 3–manifolds}
We imagine that there were some“trials to ﬁnd counterexamples to the Poincare Conjecture”
via Dehn surgery on knots.
Later Bing and Martin propose:

Property P conjecture (Bing-Martin 1960)
Let K be a nontrivial knot.
π1 (K(r)) = {1} ⇔ r = ∞Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

S 3 –Surgery Theorem
Take C∗ = {S 3 }, Gordon and Luecke prove:

Theorem (Gordon-Luecke 1989)
Let K be a nontrivial knot.
K(r) = S 3 ⇔ r = ∞Corollary (Knot complement conjecture : Tietze 1908 – Gordon-Luecke 1989)
Let K1 and K2 be knots in S 3 .
S 3 − K1 ∼= S 3 − K2 ⇔ K 1 ∼= K2

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Proof (⇐) is obvious.
(⇒) Assume that we have an orientation preserving homeomorphism
S 3 − K1 → S 3 − K2 .
By concentricity theorem we have an orientation preserving
homeomorphism E(K1 ) → E(K2 ).
∼ E(K1 ) =∼ S 1 × D2 , and hence K2 is also
• If K1 is itrivial, then E(K2 ) =trivial.
• Assume K1 and K2 are nontrivial. If h sends µ1 to µ2 , then extending h
to an orientation preserving homeomorphism h : S 3 → S 3 such thath(K1 ) = K2 .
Suppose for a contradiction that h(µ1 ) = γ2 ̸= µ2 .
Then extend h to obtain S 3 ∼= K1 (µ1 ) → K2 (γ2 ).
̸ S3,Since γ2 ̸= µ2 , Gordon-Luecke Theorem says K2 (γ2 ) ∼=a contradiction!

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Two branches of Gordon-Luecke’s S 3 –Surgery Theorem
Reformulation of Gordon-Luecke theorem.

Theorem (reformulation 1)
Let K be a nontrivial knot.
K(r) ∼= K(∞) ⇔ r = ∞⇒ cosmetic surgery slopes (discuss later)

Theorem (reformulation 2)
Let K be a knot and O a trivial knot.
K(1/n) ∼= O(1/n) = S 3 (n ̸= 0) ⇔ K ∼=O⇒ characterizing slopes (discuss later)
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Cyclic Surgery Theorem
Theorem (Culler-Gordon-Luecke-Shalen 1987)
Let K be a knot in S 3 other than a torus knot.
Assume that both K(p1 /q1 ) and K(p2 /q2 ) have cyclic fundamentalgroups.
Then the distance ∆(p1 /q1 , p2 /q2 ) = |p1 q2 − p2 q1 |≤ 1.
Hence, there are at most three such slopes.
Since K(1/0) = S 3 has the trivial, and hence cyclic fundamental group, asa consequence of this theorem we have the following:• If K is not a torus knot, then K(p/q) has a cyclic fundamental grouponly when |q| = 1, i.e. cyclic surgery is integral.
For torus knots the Property P conjecture is known to be true. Thus
noting H1 (K(p/q)) ∼= Z|p| , the Cyclic Surgery Theorem implies that• π1 (K(r)) = 1 ⇒ r = ±1.
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

The proof of the Cyclic Surgery Theorem requires“Culler-Shalen theory” + “Analysis of intersection graphs”(Culler-Shalen)(Gordon-Luecke)

Gordon-Luecke
Culler-Shalen

Culler-Shalen theory (Varieties of group representations and splittings of 3–manifolds 1983)⇒ Kabaya’s talkAnalysis of intersection graphs⇒ Litherland, Gordon-Litherland, Gordon-Luecke
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Property P Conjecture and essential lamination
Essential lamination is a generalization of“incompressible surface” and “taut foliation”.
∃ 3–manifolds that contain essential laminations but do not contain

incompressible surfaces.

(Gabai)

Gabai-Oertel 1989
If a compact orientable 3–manifold contains an essential lamination, thenits universal cover is homeomorphic to R3 .
Application to Dehn surgery.
• If K(r) has an essential lamination, then |π1 (K(r))| = ∞(strong Property P).
This leads to study of persistent lamination of knot exteriors (1990’s).
(Brittenham, Delman, Roberts, Wu....)⇒ Ito’s talkKimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Property P “Theorem”
Theorem (Kronheimer-Mrowka 2004)
Let K be a nontrivial knot.
π1 (K(r)) = {1} ⇔ r = ∞.

Theorem (reformulation)
Let K be a nontrivial knot and r ∈ Q ∪ {∞} a slope.
⟨⟨r⟩⟩ = G(K) = ⟨⟨∞⟩⟩ ⇔ r = ∞Theorem (Ito-M-Teragaito 2021)
Let K be a nontrivial knot and r, r′ ∈ Q ∪ {∞} slopes.
⟨⟨r⟩⟩ = ⟨⟨r′ ⟩⟩ ⇔ r = r′Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Hyperbolization and hyperbolic Dehn ﬁlling: Thurston 1970–Perelman 2003

|π1(M)|=8|π1(M)|<

irreducible

Seifertreducible

toroidalatoroidal

8

Classiﬁcation of 3–manifolds with ∂M = ∅ or tori

M = M1 ## Mn}prime decomposition

SeifertKimihiko Motegi (Nihon University)

hyperbolic

M i : irreducibleor
N i : atoroidal
S2 X S1

M =U Ni

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Some terminologies
A 3–manifold M is
reducible if it contains a 2–sphere which does not bound a 3–ball in M ;otherwise it is irreducible.

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

irreducible

Seifertreducible

toroidalatoroidal

8|π1(M)|=8|π1(M)|<

M = M1 ## Mn}prime decomposition

SeifertKimihiko Motegi (Nihon University)

hyperbolic

M i : irreducibleor2
N i : atoroidal
S X S1

M =U Ni

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

irreducible

Seifertreducible

toroidalatoroidal

8|π1(M)|=8|π1(M)|<

M = M1 ## Mn}prime decomposition

SeifertKimihiko Motegi (Nihon University)

hyperbolic

M i : irreducibleor2
N i : atoroidal
S X S1

M =U Ni

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Some terminologies
A 3–manifold M is
reducible if it contains a 2–sphere which does not bound a 3–ball in M ;otherwise it is irreducible.

toroidal if it contains a torus T such that i∗ : π1 (T ) → π1 (M ) is injective.
otherwise it is atoroidal.

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

irreducible

Seifertreducible

toroidalatoroidal

8|π1(M)|=8|π1(M)|<

M = M1 ## Mn}prime decomposition

SeifertKimihiko Motegi (Nihon University)

hyperbolic

M i : irreducibleor2
N i : atoroidal
S X S1

M =U Ni

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Some terminologies
A 3–manifold M is
reducible if it contains a 2–sphere which does not bound a 3–ball in M ;otherwise it is irreducible.

toroidal if it contains a torus T such that i∗ : π1 (T ) → π1 (M ) is injective.
otherwise it is atoroidal.

Seifert ﬁbered if it consists of pairwise disjoint circles (ﬁbers) in which
each ﬁber has a ﬁbered solid torus neighborhood.

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Seifert?

irreducible

reducible

toroidalatoroidal

8|π1(M)|=8|π1(M)|<

M = M1 ## Mn}prime decomposition

SeifertKimihiko Motegi (Nihon University)?

hyperbolic

M i : irreducibleor
N i : atoroidal
2
S X S1orSeifert

M =U Ni

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Hyperbolization theorem

A 3–manifold M is
hyperbolic if it admits a complete hyperbolic metric of ﬁnite volume.

Theorem (Thurston-Perelman)
Let M be an irreducible 3–manifold with ∂M = ∅ or tori.
If M is atoroidal, then it is either Seifert ﬁbered or hyperbolic.

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

|π1(M)|=8|π1(M)|<

irreducible

Seifertreducible

toroidalatoroidal

8

Classiﬁcation of 3–manifolds

M = M1 ## Mn}prime decomposition

SeifertKimihiko Motegi (Nihon University)

hyperbolic

M i : irreducibleor
N i : hyperbolic
1
2
XS
S
or
Seifert

M =U Ni

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Classiﬁcation of knots
satellite knots
C p,q (K)= f (k)torus knots

K =f (c)

f

k

c

V
(p, q)-cable of K(p, q) = (1, 2)

W(K)= f (k)kf

K =f (c)

c

T 3,4

V
Whitehead double of K

andKimihiko Motegi (Nihon University)

othersDehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

E(K)reducible
X

irreducible:

Alexander - Schonflies

toroidalatoroidal

Seiferthyperbolic

satellite knot

torus knot hyperbolic knot
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Thurston’s hyperbolic Dehn surgery
Let K be a hyperbolic knot in S 3 ,i.e. S 3 − K admits a complete hyperbolic metric of ﬁnite volume.
Q ∪ {∞} = HK ∪ EK

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Thurston’s hyperbolic Dehn surgery
Let K be a hyperbolic knot in S 3 ,i.e. S 3 − K admits a complete hyperbolic metric of ﬁnite volume.
Q ∪ {∞} = HK ∪ EK
HK = {r | K(r) is hyperbolic}, the set of hyperbolic surgery slopes
EK = {r | K(r) is not hyperbolic}, the set of “exceptional ”surgery slopes

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Thurston’s hyperbolic Dehn surgery
Let K be a hyperbolic knot in S 3 ,i.e. S 3 − K admits a complete hyperbolic metric of ﬁnite volume.
Q ∪ {∞} = HK ∪ EK
HK = {r | K(r) is hyperbolic}, the set of hyperbolic surgery slopes
EK = {r | K(r) is not hyperbolic}, the set of “exceptional ”surgery slopes

Theorem (Thurston’s hyperbolic Dehn surgery)For any hyperbolic knot K, |EK | < ∞.
K : ﬁgure-eight knot

EK = {−4, −3, −2, −1, 0, 1, 2, 3, 4, ∞}Kimihiko Motegi (Nihon University)[Thurston]

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Two directions of studies of “exceptional” surgeries onhyperbolic knots♠ Reﬁne the hyperbolic Dehn surgery theory.

|π1(M)|=8|π1(M)|<

8⇒ Ichihara’s talk
♠ Take a closer look at each exceptional surgery.

irreducible

Seifertatoroidal

reducible

toroidal

M = M1 ## Mn}prime decomposition

Seiferthyperbolic

M =U Ni

M i : irreducibleor
S2 X S1

N i : hyperbolicorSeifertReducible surgery
Seifert surgery
Toroidal surgery,and put some restrictions.
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Open problems
Cabling Conjecture (González-Acuña-Short 1986)
If K(r) is reducible, then K is a (p, q)–cable of a knot and r = pq
C p,q (K)= f (k)K =f (c)

f

k

c

V

(p, q)-cable of K

N(k)

{

(p, q) = (1, 2)

punctured lens space
1

S xD

cabling annulus

c

C S2

C

C

E(Cp,q (K))

2

E(K)

E(K)

K(p/q)#L(q, p)
Kimihiko Motegi (Nihon University)

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Diﬀerential Topology 22 19 March 2022

Conjecture (Gordon et al.)If K(r) is a Seifert ﬁber space,then r ∈ Z, or K is a torus knot or a cable of a torus knot.

Conjecture (Gordon-Wu 2008)
A hyperbolic knot K has at most three toroidal surgeries.

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

{Lens space surgeries} ⊂ {Seifert ﬁbered surgery}
Let K be a knot contained in a genus 2 Heegaard surface F which splits
S 3 into two genus 2 handlebodies H and H ′ , i.e. S 3 = H ∪F H ′ .
H’
F
K
H

Performing Dehn surgery on K along the surface slope m,
2-handle
H’F
surface slopeFH

2-handle

we obtain a 3-manifold
K(m) = H[K] ∪ H ′ [K]Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

{Lens space surgeries} ⊂ {Seifert ﬁbered surgery}
Let K be a knot contained in a genus 2 Heegaard surface F which splits
S 3 into two genus 2 handlebodies H and H ′ , i.e. S 3 = H ∪F H ′ .
H’
F
K
H

Performing Dehn surgery on K along the surface slope m,
2-handle
H’F
surface slopeFH

2-handle

we obtain a 3-manifold
K(m) = H[K] ∪ H ′ [K] = (S 1 × D2 ) ∪ (S 1 × D2 ),which is a lens spaces.
This construction is called Berge’s primitive/primitive construction.
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Berge introduced “Berge’s lens space surgeries”= 12 classes of primitive/primitive knots (positions)

Berge’s Conjecture
Every lens space surgery is a Berge’s lens space surgery.
(i) Every lens space surgery arises from primitive/primitive construction,(ii) all primitive/primitive knots belong to Berge’s list. (Greene 2010)⇒ Teragaito’s talk & Tange’s talk & Yamada’s talkKimihiko Motegi (Nihon University)

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Diﬀerential Topology 22 19 March 2022

Then by performing Dehn surgery on K along the surface slope m,
we obtain a 3-manifold
K(m) = H[K] ∪ H ′ [K] = (S 1 × D2 ) ∪ (S 1 × D2 ),which is a lens spaces.
2-handle
H’F
surface slopeFH

2-handle

This construction is called Berge’s primitive/primitive construction.
Berge introduced ”Berge’s lens space surgeries” = 12 classes ofprimitive/primitive knots (positions)

Berge’s Conjecture
Every lens space surgery is a Berge’s lens space surgery.
(i) Every lens space surgery arises from primitive/primitive construction,(ii) all primitive/primitive knots belong to Berge’s list. (Greene 2010)⇒ Teragaito’s talk & Tange’s talk & Yamada’s talkKimihiko Motegi (Nihon University)

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Diﬀerential Topology 22 19 March 2022

What comes next?

Reducible surgeries, toroidal surgeries and Seifert surgeries are“exceptional” with respect to Thurston’ hyperbolic Dehn surgery.

L-space surgeries, generalization of lens space surgeries, are “exceptional”
with respect to in Heegaard Floer homology theory.

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

L–spaces

M : rational homology 3–sphere

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

L–spaces

M : rational homology 3–sphered (M ) : Heegaard Floer homology with coeﬃcients in Z2
HF

Kimihiko Motegi (Nihon University)

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Diﬀerential Topology 22 19 March 2022

L–spaces

M : rational homology 3–sphered (M ) : Heegaard Floer homology with coeﬃcients in Z2
HF
d (M )≥|H1 (M ; Z)|
dimZ2 HF

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

L–spaces

M : rational homology 3–sphered (M ) : Heegaard Floer homology with coeﬃcients in Z2
HF
d (M )≥|H1 (M ; Z)|
dimZ2 HF
d (M )=|H1 (M ; Z)|.
M is an L–space if “=” holds, i.e. dimZ2 HF

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

L–spaces

M : rational homology 3–sphered (M ) : Heegaard Floer homology with coeﬃcients in Z2
HF
d (M )≥|H1 (M ; Z)|
dimZ2 HF
d (M )=|H1 (M ; Z)|.
M is an L–space if “=” holds, i.e. dimZ2 HF
ExampleLens spaces (̸= S 2 × S 1 ), more generally, 3–manifolds with S 3 –geometry
are L–spaces.

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Diﬀerential Topology 22 19 March 2022

L–space knots
A knot K is called an L–space knot if it admits a nontrivial, L–space
surgery.

Kimihiko Motegi (Nihon University)

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Diﬀerential Topology 22 19 March 2022

L–space knots
A knot K is called an L–space knot if it admits a nontrivial, L–space
surgery.
• K: a nontrivial L–space knot

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Diﬀerential Topology 22 19 March 2022

L–space knots
A knot K is called an L–space knot if it admits a nontrivial, L–space
surgery.
• K: a nontrivial L–space knot
K(r) is an L–space ⇔ r ≥ 2g(K) − 1 or r ≤ −2g(K) + 1.
(Ozsváth-Szabó 2011)
L-space surgeries

02g(K)-1

Q

L-space surgeries-2g(K)+1

0QKnot Floer homology, surgery formula ⇒ Sato’s talkKimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

L–space knots
A knot K is called an L–space knot if it admits a nontrivial, L–space
surgery.
• K: a nontrivial L–space knot
K(r) is an L–space ⇔ r ≥ 2g(K) − 1 or r ≤ −2g(K) + 1.
(Ozsváth-Szabó 2011)
L-space surgeries

02g(K)-1

Q

L-space surgeries-2g(K)+1

0QKnot Floer homology, surgery formula ⇒ Sato’s talk• If K is hyperbolic L-space knot, it produces inﬁnitely many hyperbolic
L–spaces by Dehn surgery. (Thurston’s hyperbolic Dehn surgery theorem)
Kimihiko Motegi (Nihon University)

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Diﬀerential Topology 22 19 March 2022

Constraints for L–space knots• For any L–space knot K,
the non-zero coeﬃcients of the Alexander polynomial of K are ±1 andalternate in sign.
(Ozsváth-Szabó 2005)⇒ Teragaito’s talkKimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Constraints for L–space knots• For any L–space knot K,
the non-zero coeﬃcients of the Alexander polynomial of K are ±1 andalternate in sign.
(Ozsváth-Szabó 2005)⇒ Teragaito’s talk
• An L-space knot is a ﬁbered knot,
i.e. its exterior is a surface bundle over S 1 .

(Ni 2007)

Corollary (Ni)A knot with lens space surgery, more generally ﬁnite surgery is ﬁbered.

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

• L-space knot is prime (1-string prime).

（Krcarovich 2015）(It should be compare with: Connected sum of L-spaces is an L-space!)• An L-space knot has no essential Conway sphere, i.e. 2-string prime.
(Lidman-Moore-Zibrowius 2020)[Hanselman-Rasmussen-Watson;
Bordered Floer homology for manifolds with torus boundary via immersed curves] (Ito’s talk)enables us to prove:• Any satellite L-space knot has an
L-space knot companion and L-space pattern knot
andbraided pattern.

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey（Hom 2016）(Baker-M 2019)

Diﬀerential Topology 22 19 March 2022

Surjectivity (realization) problem of Dehn surgery
Is D : {knots in S 3 } → {closed 3–manifolds with weight one π1 } surjective?
Boyer-Lines (1990) found (ﬁrst) examples of weight one 3–manifolds whichcannot be obtained by Dehn surgery on knots.
These 3–manifolds are Seifert ﬁber spaces.
Auckly (1993) gave an example of a hyperbolic homology 3–sphere M which is
not obtained by Dehn surgery on a knot, but it is not known that π1 (M ) has
weight one.
Hoﬀman and Walsh (2015) give inﬁnite family of hyperbolic 3–manifolds withweight one fundamental group none of which cannot be obtained by surgery on aknot in S 3 .
Hom, Karakurt and Lidman (2016) have given an inﬁnite family of Seifert
homology 3–spheres each of which has weight one, but not obtained by Dehn
surgery on a knot in S 3 .
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Injectivity problem of Dehn surgery

D : {knots in S 3 } × Q → {oriented closed 3–manifolds}
D(K, r) = K(r)Then ﬁxing a knot K ⊂ S 3 or a surgery slope r ∈ Q, we may consider two
kinds of injectivity of D.

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Fix a knot K in S 3 . Then is the map
DK : Q → {oriented closed 3–manifolds}
injective?
In other words, if K(r) = K(r′ ), then r = r′ ?

Fix a rational number r ∈ Q. Then is the map
Dr : {knots in S 3 } → {oriented closed 3–manifolds}
injective?
In other words, if K(r) = K ′ (r), then K = K ′ ?

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Injectivity of the map DK
Is the map
DK : Q → {oriented closed 3–manifolds}
DK (r) = K(r)injective?

Cosmetic surgery slope
For a given knot K we say that r ∈ Q is a cosmetic surgery slope for K
if ∃ r′ ̸= r s.t. K(r′ ) ∼= K(r).
Since H1 (K(p/q)) ∼= Zp (p ≥ 0), if K(p/q) ∼= K(p′ /q ′ ), then p = p′ .
Let O be the trivial knot in S 3 .
Then O(1/n) ∼= S 3 for any integer n, and thus DO is not injective.
1/n is a cosmetic surgery slope for “trivial knot”.
(Actually there is an orientation preserving homeomorphism of E(O) = S 1 × D2
sending 1/n to 1/0.)
Kimihiko Motegi (Nihon University)

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Diﬀerential Topology 22 19 March 2022

Theorem (Gordon-Luecke: reformulation 1)
Let K be a nontrivial knot.
∞ is not a cosmetic surgery slope for K, i.e. K(r) ∼= K(∞) ⇔ r = ∞.

Cosmetic surgery conjecture
Any nontrivial knot in S 3 does not have a cosmetic surgery slope, i.e.
DK : Q → {oriented closed 3–manifolds}
DK (r) = K(r)is injective.
If M = K(r1 ) ∼= K(r2 ), then some nontrivial surgery on
Kr∗1 ⊂ M = K(r1 ) converts M = K(r1 ) into M = K(r2 ) = K(r1 ).
This is the original meaning of “cosmetic” surgery.
Kimihiko Motegi (Nihon University)

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Diﬀerential Topology 22 19 March 2022

DK : Q → {unorineted closed 3–manifolds} is not injective.
Indeed, Mathieu observed T3,2 (9) ∼= −T3,2 (9/2), where −M denotes M
with the opposite orientation.

Theorem (Ni-Wu 2015)
Assume that K is nontrivial and K(p/q) ∼= K(p/q ′ ) for distinct slopes′
p/q, p/q ∈ Q ∪ {∞}.
Then q = −q ′ and q 2 ≡ −1 (mod p).

Theorem (Hanselman 2019)
Assume that K(r) ∼= K(r′ ) for distinct slopes r, r′ ∈ Q. Then
1
2{r, r′ } = {±2} or {±1/q}
If {r, r′ } = {±1/q}, then there are at most ﬁnitely many q; thenumber of q is depending on q and decidable.

⇒ Ito’s talkKimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Injectivity of the map Dr
Is the map
Dr : K → {oriented closed 3–manifolds}
Dr (K) = K(r)injective?

ExampleLickorish provided the ﬁrst example of distinct knots K1 and K2 which
satisfy K1 (1) ∼= K2 (1), answering Kirby’s question (1976).
1

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1

Diﬀerential Topology 22 19 March 2022

Characterizing slope
A slope r ∈ Q is a characterizing slope for a knot K
if whenever K ′ (r) ∼= K(r), then K ′ is isotopic to K.

Theorem (Gordon-Luecke: reformulation 2)
Let K be a knot and O a trivial knot.
K(1/n) ∼= O(1/n) (n ̸= 0) ⇔ K ∼=OThis means that
1/n (n ̸= 0) is a characterizing slope for the trivial knot.
• Gordon had conjectured thatevery r ∈ Q is a characterizing slope for the trivial knot.
about ten years before he proved knot complement conjecture with Luecke.
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Theorem (Kronheimer-Mrowka-Ozsváth-Szabó 2007)
For the trivial knot every r ∈ Q is a characterizing slope.
This answers Gordon’s conjecture aﬃrmatively.
Apply this theorem to prove “projective space conjecture”.

Theorem (projective space conjecture)
If K(r) is the projective space RP 3 , then K is the trivial knot.
Proof. Assume that K(m/n) ∼= RP 3 . Then m = 2 for homological reason.
If K is a nontrivial torus knot Tp,q , then we must have |pqn − 2| = 1,
which is impossible because |pq| ≥ 6.
So K is not a torus knot, and Cyclic Surgery Theorem shows that |n| = 1.
K(2) ∼□= RP 3 ∼= O(2) implies K ∼= O by the above theorem.
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Theorem (Kronheimer-Mrowka-Ozsváth-Szabó 2007)
For the trivial knot every r ∈ Q is a characterizing slope.
This theorem is generalized to

Theorem (Ozsváth-Szabó 2019)
Let K be a trefoil knot or the ﬁgure-eight knot.
Then every r ∈ Q is a characterizing slope for K.

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Theorem (Ni-Zhang 2014)
Let K be a torus knot Tp,q (p > q ≥ 2).
If r > 30(p2 − 1)(q 2 − 1)/67, then r is a characterizing slope for K.

Question (Ni-Zhang 2014) Let K be a hyperbolic knot.
Then is p/q a characterizing slope for K if |p| + |q| is suﬃciently large?
On the other hand,• For given integer m,∃

hyperbolic knots K, K ′ s.t. K(m) ∼= K ′ (m).
(Kawauchi)

Kimihiko Motegi (Nihon University)

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Diﬀerential Topology 22 19 March 2022

Non-characterizing slopes

Theorem (Baker-M 2018)
1

There exists a hyperbolic knot for which every integer is a
non-characterizing slope.

2

For a given nontrivial knot k, there exists a satellite knot withcompanion knot k for which every integer is a non-characterizing
slope.

This is based on Gompf-Miyazaki’s “surgery dual” (1995).

Kimihiko Motegi (Nihon University)

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Diﬀerential Topology 22 19 March 2022

Existence of characterizing slopes
Theorem (Lackenby 2019, McCoy 2019)1

Every hyperbolic knot has a characterizing slope.

2

Any torus knot Tr,s with r, s > 1 has only ﬁnitely many
non-characterizing slopes which are not negative integers.

3(Lackenby)(McCoy)Any hyperbolic knot can have only ﬁnitely many non-characterizing
slopes with q ≥ 3(McCoy)Conjecture (Universal characterizing slope)
There exist a universal characterizing slope r, i,e, the map
Dr : K → {oriented closed 3–manifolds}
Dr (K) = K(r)injective. In other words, r is a characterizing slope for all knots.
Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Let m ∈ Z be a non-characterizing slope for a knot K.
Then by deﬁnition, there is a knot K ′ which is not isotopic to K and
K ′ (m) ∼= K(m).
It is interesting to ask if there are inﬁnitely knots Ki (i = 1, 2, . . . ) suchthat
K1 (m) ∼= K2 (m) ∼= ··· ∼
= Kn (m) ∼= ···(Kirby)Osoinach answered this question in the positive when m = 0.
Some extensions were given by Teragaito, Kouno.

Theorem ( Abe, Jong, Luecke and Osoinach 2015)
For every integer n, there exist inﬁnitely many knots K1 , K2 . . . such that
∼ ···.
K1 (n) ∼= K2 (n) =Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Diﬀerential Topology 22 19 March 2022

Q

cosmetic surgery slope

cosmetic surgery slope

K

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

knots

Diﬀerential Topology 22 19 March 2022

Qrcharacterizing slope

K

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Ki

knots

Diﬀerential Topology 22 19 March 2022

Qrcharacterizing slope

K

Kimihiko Motegi (Nihon University)

Dehn surgery on knots：survey

Ki

Kj

knots

Diﬀerential Topology 22 19 March 2022

Qruniversal
characterizing slope

K (r) = K' (r)K

Kimihiko Motegi (Nihon University)

for all pair of knots K and K'

Ki

Dehn surgery on knots：survey

Kj

knots

Diﬀerential Topology 22 19 March 2022

L-space surgery and left-orderable surgery
Left orderable
A nontrivial group G is said to be left-orderable
if there exists a strict total ordering < that is invariant under multiplicationfrom the left, i.e. if g < h, then f g < f h for any f, g, h ∈ G.
• A left-orderable group has no torsion.
Let G be an left-orderable group and g a nontrivial element. Assume that 1 < g.
Then 1 < g = g · 1 < g · g = g 2 < g 3 < · · · < g n for any n ≥ 2.

So if K(r) is a lens space, π1 (K(r)) is not left-orderable.

L-space conjecture (Boyer, Gordon and Watson 2013)
M : irreducible, rational homology 3–sphere
M is an L-space⇔
π1 (M ) is not left-orderable.
Kimihiko Motegi (Nihon University)

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Diﬀerential Topology 22 19 March 2022

A 3–manifold with geometric structure other than hyperbolic structure
satisﬁes the L-space conjecture.
Furthermore, many hyperbolic 3–manifolds are known to satisfy the
conjecture.
(Boyer, Gordon and Watson 2013)For any knot K ⊂ S 3 , since G(K) = π1 (E(K)) admits an epimorphism
onto the inﬁnite cyclic group Z via its abelianization, G(K) is
left-orderable.
(Howie-Short 1985, Boyer-Rolfsen-Wiest 2005)
However, the fundamental group of K(r) may not be left-orderable.

SL (K) = {r ∈ Q | K(r) is an L-space}
SLO (K) = {r ∈ Q | π1 (K(r)) is left-orderable}

Conjecture
Let K be a knot in S 3 . Assume that K is not a cable of a nontrivial knot.
Then SL (K) ∪ SLO (K) = Q and SL (K) ∩ SLO (K) = ∅.
Kimihiko Motegi (Nihon University)

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Diﬀerential Topology 22 19 March 2022

Torus knot
For (p, q)–torus knot Tp,q (p > q ≥ 2),
SLO (Tp,q ) = (−∞, pq − p − q) ∩ Q, andSL (Tp,q ) = [pq − p − q, ∞) ∩ Q.
• There exist inﬁnitely many hyperbolic knots K such that
SLO (K) = Q and SL (K) = ∅.
(M-Teragaito 2014)

Kimihiko Motegi (Nihon University)

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Diﬀerential Topology 22 19 March 2022