Lens space fillings of once-punctured torus bundles
Kenneth L. [email protected]
Georgia Institute of TechnologyAtlanta, Georgia, USA
VII Reunión Conjunta AMS-SMM26 de mayo de 2007
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 1 / 26
Lens space fillings of OPT–bundles: ∆ = 1 and ∆ > 1
TheoremM is an OPT–bundle admitting a lens space filling of distance ∆
m
M ∼= F × I/φ with φ = τnx τ2
y τmx τ−1
y for some m, n ∈ Z.
Moreover, ∆ = 1 with the following exceptions:
φ = τxτy and ∆ = | − 6k + 1| for k ∈ Z
φ = τ2x τy and ∆ = | − 4k + 1| for k ∈ Z
φ = τ3x τy and ∆ = | − 3k + 1| for k ∈ Z
φ = τ4x τy and ∆ = 1, 3
φ = τ5x τy and ∆ = 1, 2
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 2 / 26
Lens space fillings of OPT–bundles: ∆ = 1 and ∆ > 1
TheoremM is an OPT–bundle admitting a lens space filling of distance ∆
m
M ∼= F × I/φ with φ = τnx τ2
y τmx τ−1
y for some m, n ∈ Z.
Moreover, ∆ = 1 with the following exceptions:
φ = τxτy and ∆ = | − 6k + 1| for k ∈ Z
φ = τ2x τy and ∆ = | − 4k + 1| for k ∈ Z
φ = τ3x τy and ∆ = | − 3k + 1| for k ∈ Z
φ = τ4x τy and ∆ = 1, 3
φ = τ5x τy and ∆ = 1, 2
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 2 / 26
Theorem (Burde & Zieschang, González-Acuña)
In S3 there are only two genus one fibered knots.(Up to homeomorphism)
The Trefoil The Figure Eight
(. . . but infinitely many genus one knots.)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 3 / 26
Theorem (Burde & Zieschang, González-Acuña)
In S3 there are only two genus one fibered knots.(Up to homeomorphism)
The Trefoil The Figure Eight
(. . . but infinitely many genus one knots.)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 3 / 26
Theorem (Burde & Zieschang, González-Acuña)
In S3 there are only two genus one fibered knots.(Up to homeomorphism)
The Trefoil The Figure Eight
(. . . but infinitely many genus one knots.)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 3 / 26
Theorem (Burde & Zieschang, González-Acuña)
In S3 there are only two genus one fibered knots.(Up to homeomorphism)
The Trefoil The Figure Eight
(. . . but infinitely many genus one knots.)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 3 / 26
Definition
A knot K ⊂ M̂closed
is an OPT–knot if there is an essential
Once-Punctured Torus F
properly embedded in its exterior M = M̂ − N(K ).
DefinitionLet µ be the meridian of K . Then ∆ = ∆(µ, ∂F ) = |µ · ∂F | is
∆ = 8
the distance of µ and ∂F ,
the distance of the filling ofthe exterior M by N(K ), and
the order of K in H1(M̂).
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 4 / 26
Definition
A knot K ⊂ M̂closed
is an OPT–knot if there is an essential
Once-Punctured Torus F
properly embedded in its exterior M = M̂ − N(K ).
DefinitionLet µ be the meridian of K . Then ∆ = ∆(µ, ∂F ) = |µ · ∂F | is
∆ = 8
the distance of µ and ∂F ,
the distance of the filling ofthe exterior M by N(K ), and
the order of K in H1(M̂).
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 4 / 26
Definition
A knot K ⊂ M̂closed
is an OPT–knot if there is an essential
Once-Punctured Torus F
properly embedded in its exterior M = M̂ − N(K ).
DefinitionLet µ be the meridian of K . Then ∆ = ∆(µ, ∂F ) = |µ · ∂F | is
∆ = 8
the distance of µ and ∂F ,
the distance of the filling ofthe exterior M by N(K ), and
the order of K in H1(M̂).
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 4 / 26
Definition
A knot K ⊂ M̂closed
is an OPT–knot if there is an essential
Once-Punctured Torus F
properly embedded in its exterior M = M̂ − N(K ).
DefinitionLet µ be the meridian of K . Then ∆ = ∆(µ, ∂F ) = |µ · ∂F | is
∆ = 8
the distance of µ and ∂F ,
the distance of the filling ofthe exterior M by N(K ), and
the order of K in H1(M̂).
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 4 / 26
Definition
A knot K ⊂ M̂closed
is an OPT–knot if there is an essential
Once-Punctured Torus F
properly embedded in its exterior M = M̂ − N(K ).
DefinitionLet µ be the meridian of K . Then ∆ = ∆(µ, ∂F ) = |µ · ∂F | is
∆ = 8
the distance of µ and ∂F ,
the distance of the filling ofthe exterior M by N(K ), and
the order of K in H1(M̂).
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 4 / 26
Definition
A knot K ⊂ M̂closed
is an OPT–knot if there is an essential
Once-Punctured Torus F
properly embedded in its exterior M = M̂ − N(K ).
DefinitionLet µ be the meridian of K . Then ∆ = ∆(µ, ∂F ) = |µ · ∂F | is
∆ = 8
the distance of µ and ∂F ,
the distance of the filling ofthe exterior M by N(K ), and
the order of K in H1(M̂).
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 4 / 26
Definition
A knot K ⊂ M̂closed
is an OPT–knot if there is an essential
Once-Punctured Torus F
properly embedded in its exterior M = M̂ − N(K ).
DefinitionLet µ be the meridian of K . Then ∆ = ∆(µ, ∂F ) = |µ · ∂F | is
∆ = 8
the distance of µ and ∂F ,
the distance of the filling ofthe exterior M by N(K ), and
the order of [K ] in H1(M̂).
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 4 / 26
∆ = 1
=⇒ K is null homologous
∆[K ] = [K ] = 0 ∈ H1(M̂; Z)
ZOPT–knot
∆ > 1
=⇒ K is rationally nullhomologous
∆[K ] = 0 ∈ H1(M̂; Z)
[K ] = 0 ∈ H1(M̂; Q)
QOPT–knot
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 5 / 26
∆ = 1
=⇒ K is null homologous
∆[K ] = [K ] = 0 ∈ H1(M̂; Z)
ZOPT–knot
∆ > 1
=⇒ K is rationally nullhomologous
∆[K ] = 0 ∈ H1(M̂; Z)
[K ] = 0 ∈ H1(M̂; Q)
QOPT–knot
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 5 / 26
DefinitionIf the exterior of K is a once-punctured torus bundle (OPT–bundle )
F × I/φ
then K is a OPT–fibered knot :ZOPT–fibered knot if ∆ = 1 (genus one fibered knot)QOPT–fibered knot if ∆ > 1
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 6 / 26
DefinitionIf the exterior of K is a once-punctured torus bundle (OPT–bundle )
F × I/φ
then K is a OPT–fibered knot :ZOPT–fibered knot if ∆ = 1 (genus one fibered knot)QOPT–fibered knot if ∆ > 1
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 6 / 26
DefinitionIf the exterior of K is a once-punctured torus bundle (OPT–bundle )
F × I/φ
then K is a OPT–fibered knot :ZOPT–fibered knot if ∆ = 1 (genus one fibered knot)QOPT–fibered knot if ∆ > 1
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 6 / 26
DefinitionIf the exterior of K is a once-punctured torus bundle (OPT–bundle )
F × I/φ
then K is a OPT–fibered knot :ZOPT–fibered knot if ∆ = 1 (genus one fibered knot)QOPT–fibered knot if ∆ > 1
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 6 / 26
DefinitionIf the exterior of K is a once-punctured torus bundle (OPT–bundle )
F × I/φ
then K is a OPT–fibered knot :ZOPT–fibered knot if ∆ = 1 (genus one fibered knot)QOPT–fibered knot if ∆ > 1
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 6 / 26
DefinitionIf the exterior of K is a once-punctured torus bundle (OPT–bundle )
F × I/φ
then K is a OPT–fibered knot :ZOPT–fibered knot if ∆ = 1 (genus one fibered knot)QOPT–fibered knot if ∆ > 1
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 6 / 26
The monodromy φ is acomposition of Dehn twistsalong basis curves x , y on F .
τx : 7→
τy : 7→
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 7 / 26
The monodromy φ is acomposition of Dehn twistsalong basis curves x , y on F .
τx : 7→
τy : 7→
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 7 / 26
The monodromy φ is acomposition of Dehn twistsalong basis curves x , y on F .
τx : 7→
τy : 7→
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 7 / 26
Example
The Trefoil
φ = τxτy
The Figure Eight
φ = τ−1x τy .
These are the two OPT–bundles that may be filled to produce S3.∆ = 1.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 8 / 26
Example
The Trefoil
φ = τxτy
The Figure Eight
φ = τ−1x τy .
These are the two OPT–bundles that may be filled to produce S3.∆ = 1.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 8 / 26
Definition
The Lens spaces L(p, q) aretwo solid tori glued togetheralong their boundaries.
To form L(p, q), the meridian ofone becomes a (p, q) curve onthe other.
Example
S3 ∼= L(1, n) for n ∈ Z RP2 ∼= L(2, 1) S1 × S2 ∼= L(0, 1)
L(p, q) ∼= L(p′, q′) ⇐⇒ |p′| = |p| and q′ ≡ ±q±1 mod p
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 9 / 26
ProblemExtend classification of OPT–fibered knots to lens spaces.
Morimoto began the classification of ZOPT–fibered knots in lensspaces up to homeomorphism:
L(m, 1) contains at least two if m > 0,exactly two if m ∈ {1, 2, 3, 5, 19},
L(4, 1) contains exactly three,L(0, 1), L(5, 2), and L(19, 3) contain exactly one,L(19, 2), L(19, 4), and L(19, 7) contain none.
=⇒ determined some OPT–bundles with lens space fillings.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 10 / 26
ProblemExtend classification of OPT–fibered knots to lens spaces.
Morimoto began the classification of ZOPT–fibered knots in lensspaces up to homeomorphism:
L(m, 1) contains at least two if m > 0,exactly two if m ∈ {1, 2, 3, 5, 19},
L(4, 1) contains exactly three,L(0, 1), L(5, 2), and L(19, 3) contain exactly one,L(19, 2), L(19, 4), and L(19, 7) contain none.
=⇒ determined some OPT–bundles with lens space fillings.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 10 / 26
ProblemExtend classification of OPT–fibered knots to lens spaces.
Morimoto began the classification of ZOPT–fibered knots in lensspaces up to homeomorphism:
L(m, 1) contains at least two if m > 0,exactly two if m ∈ {1, 2, 3, 5, 19},
L(4, 1) contains exactly three,L(0, 1), L(5, 2), and L(19, 3) contain exactly one,L(19, 2), L(19, 4), and L(19, 7) contain none.
=⇒ determined some OPT–bundles with lens space fillings.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 10 / 26
QuestionWhat are the OPT–fibered knots in lens spaces?What OPT–bundles may be filled to produces a lens space?
ApproachFor ∆ = 1, we classify ZOPT–fibered knots in lens spaces.
(. . . infinitely many ZOPT–knots.)
For ∆ > 1, we classify QOPT–knots in lens spacesand then observe which are fibered.
(. . . infinitely many QOPT–knots?)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 11 / 26
QuestionWhat are the OPT–fibered knots in lens spaces?What OPT–bundles may be filled to produces a lens space?
ApproachFor ∆ = 1, we classify ZOPT–fibered knots in lens spaces.
(. . . infinitely many ZOPT–knots.)
For ∆ > 1, we classify QOPT–knots in lens spacesand then observe which are fibered.
(. . . infinitely many QOPT–knots?)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 11 / 26
QuestionWhat are the OPT–fibered knots in lens spaces?What OPT–bundles may be filled to produces a lens space?
ApproachFor ∆ = 1, we classify ZOPT–fibered knots in lens spaces.
(. . . infinitely many ZOPT–knots.)
For ∆ > 1, we classify QOPT–knots in lens spacesand then observe which are fibered.
(. . . infinitely many QOPT–knots?)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 11 / 26
QuestionWhat are the OPT–fibered knots in lens spaces?What OPT–bundles may be filled to produces a lens space?
ApproachFor ∆ = 1, we classify ZOPT–fibered knots in lens spaces.
(. . . infinitely many ZOPT–knots.)
For ∆ > 1, we classify QOPT–knots in lens spacesand then observe which are fibered.
(. . . infinitely many QOPT–knots?)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 11 / 26
QuestionWhat are the OPT–fibered knots in lens spaces?What OPT–bundles may be filled to produces a lens space?
ApproachFor ∆ = 1, we classify ZOPT–fibered knots in lens spaces.
(. . . infinitely many ZOPT–knots.)
For ∆ > 1, we classify QOPT–knots in lens spacesand then observe which are fibered.
(. . . infinitely many QOPT–knots?)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 11 / 26
QuestionWhat are the OPT–fibered knots in lens spaces?What OPT–bundles may be filled to produces a lens space?
ApproachFor ∆ = 1, we classify ZOPT–fibered knots in lens spaces.
(. . . infinitely many ZOPT–knots.)
For ∆ > 1, we classify QOPT–knots in lens spacesand then observe which are fibered.
(. . . infinitely many QOPT–knots?)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 11 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
Idea
ZOPT–fibered knots correspond to braid axes of closed 3–braids in S3.
Example (The Figure Eight)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 12 / 26
∆ = 1
1–1 Correspondence
A closed 3–manifold M̂with ZOPT–fibered knot K
(M̂, K )↔
A link L in S3
with double branched cover M̂and an axis (unknot) A
that presents L as closed3–braid(L, A)
↔
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 13 / 26
∆ = 1
1–1 Correspondence
A closed 3–manifold M̂with ZOPT–fibered knot K
(M̂, K )↔
A link L in S3
with double branched cover M̂and an axis (unknot) A
that presents L as closed3–braid(L, A)
↔
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 13 / 26
∆ = 1
1–1 Correspondence
A closed 3–manifold M̂with ZOPT–fibered knot K
(M̂, K )↔
A link L in S3
with double branched cover M̂and an axis (unknot) A
that presents L as closed3–braid(L, A)
↔
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 13 / 26
∆ = 1
For lens spaces M̂ = L(p, q)
[Hodgson & Rubinstein] Lens spacesare only double branched covers of2–bridge links.
[Murasugi, Stoimenow] The 2–bridgelinks of braid index 3 or less are thoseshown.
[Birman & Menasco] The braid axes oflinks with braid index at most 3 areclassified.
When considered up tohomeomorphisms, we have the shown3–braid presentations of 2–bridge links.
All have the form σnx σ2
yσmx σ−1
y
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 14 / 26
∆ = 1
For lens spaces M̂ = L(p, q)
[Hodgson & Rubinstein] Lens spacesare only double branched covers of2–bridge links.
[Murasugi, Stoimenow] The 2–bridgelinks of braid index 3 or less are thoseshown.
[Birman & Menasco] The braid axes oflinks with braid index at most 3 areclassified.
When considered up tohomeomorphisms, we have the shown3–braid presentations of 2–bridge links.
All have the form σnx σ2
yσmx σ−1
y
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 14 / 26
∆ = 1
For lens spaces M̂ = L(p, q)
[Hodgson & Rubinstein] Lens spacesare only double branched covers of2–bridge links.
[Murasugi, Stoimenow] The 2–bridgelinks of braid index 3 or less are thoseshown.
[Birman & Menasco] The braid axes oflinks with braid index at most 3 areclassified.
When considered up tohomeomorphisms, we have the shown3–braid presentations of 2–bridge links.
All have the form σnx σ2
yσmx σ−1
y
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 14 / 26
∆ = 1
For lens spaces M̂ = L(p, q)
[Hodgson & Rubinstein] Lens spacesare only double branched covers of2–bridge links.
[Murasugi, Stoimenow] The 2–bridgelinks of braid index 3 or less are thoseshown.
[Birman & Menasco] The braid axes oflinks with braid index at most 3 areclassified.
When considered up tohomeomorphisms, we have the shown3–braid presentations of 2–bridge links.
All have the form σnx σ2
yσmx σ−1
y
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 14 / 26
∆ = 1
For lens spaces M̂ = L(p, q)
[Hodgson & Rubinstein] Lens spacesare only double branched covers of2–bridge links.
[Murasugi, Stoimenow] The 2–bridgelinks of braid index 3 or less are thoseshown.
[Birman & Menasco] The braid axes oflinks with braid index at most 3 areclassified.
When considered up tohomeomorphisms, we have the shown3–braid presentations of 2–bridge links.
All have the form σnx σ2
yσmx σ−1
y
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 14 / 26
∆ = 1
For lens spaces M̂ = L(p, q)
[Hodgson & Rubinstein] Lens spacesare only double branched covers of2–bridge links.
[Murasugi, Stoimenow] The 2–bridgelinks of braid index 3 or less are thoseshown.
[Birman & Menasco] The braid axes oflinks with braid index at most 3 areclassified.
When considered up tohomeomorphisms, we have the shown3–braid presentations of 2–bridge links.
All have the form σnx σ2
yσmx σ−1
y
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 14 / 26
∆ = 1
TheoremUp to homeomorphism, the lens space L(p, q) contains exactly
three ZOPT–fibered knots ⇐⇒ L(p, q) ∼= L(4, 1),
two ZOPT–fibered knots ⇐⇒ L(p, q) ∼= L(r , 1) for r > 0, 6= 4,
one ZOPT–fibered knot ⇐⇒ L(p, q) ∼= L(r , s) for either r = 0or for 0 < s < r where either
r = 2mn + m + n and s = 2n + 1 for some integers n, m > 1, orr = 2mn +m +n +1 and s = 2n +1 for some integers n, m > 0, and
no ZOPT–fibered knots otherwise.
TheoremThe OPT–bundle exterior of any ZOPT–fibered knot in a lens spacehas monodromy of the form
φ = τnx τ2
y τmx τ−1
y .
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 15 / 26
∆ = 1
TheoremUp to homeomorphism, the lens space L(p, q) contains exactly
three ZOPT–fibered knots ⇐⇒ L(p, q) ∼= L(4, 1),
two ZOPT–fibered knots ⇐⇒ L(p, q) ∼= L(r , 1) for r > 0, 6= 4,
one ZOPT–fibered knot ⇐⇒ L(p, q) ∼= L(r , s) for either r = 0or for 0 < s < r where either
r = 2mn + m + n and s = 2n + 1 for some integers n, m > 1, orr = 2mn +m +n +1 and s = 2n +1 for some integers n, m > 0, and
no ZOPT–fibered knots otherwise.
TheoremThe OPT–bundle exterior of any ZOPT–fibered knot in a lens spacehas monodromy of the form
φ = τnx τ2
y τmx τ−1
y .
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 15 / 26
∆ > 1
For ∆ > 1, the preceding argument does not work so well.
Instead we examine how a QOPT–knot K with OPTF interacts with aHeegaard torus T̂ of the lens space L(p, q).
If K is a torus knot (scc on T̂ ), then straightforward.Assume K is not a torus knot, then:
Put K into thin position with respect to T̂ .Show that K must be 1–bridge with respect to T̂ .Show that ∆ = 2 or ∆ = 3.Analyze resulting possible configurations.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 16 / 26
∆ > 1
For ∆ > 1, the preceding argument does not work so well.
Instead we examine how a QOPT–knot K with OPTF interacts with aHeegaard torus T̂ of the lens space L(p, q).
If K is a torus knot (scc on T̂ ), then straightforward.Assume K is not a torus knot, then:
Put K into thin position with respect to T̂ .Show that K must be 1–bridge with respect to T̂ .Show that ∆ = 2 or ∆ = 3.Analyze resulting possible configurations.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 16 / 26
∆ > 1
For ∆ > 1, the preceding argument does not work so well.
Instead we examine how a QOPT–knot K with OPTF interacts with aHeegaard torus T̂ of the lens space L(p, q).
If K is a torus knot (scc on T̂ ), then straightforward.Assume K is not a torus knot, then:
Put K into thin position with respect to T̂ .Show that K must be 1–bridge with respect to T̂ .Show that ∆ = 2 or ∆ = 3.Analyze resulting possible configurations.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 16 / 26
∆ > 1
For ∆ > 1, the preceding argument does not work so well.
Instead we examine how a QOPT–knot K with OPTF interacts with aHeegaard torus T̂ of the lens space L(p, q).
If K is a torus knot (scc on T̂ ), then straightforward.Assume K is not a torus knot, then:
Put K into thin position with respect to T̂ .Show that K must be 1–bridge with respect to T̂ .Show that ∆ = 2 or ∆ = 3.Analyze resulting possible configurations.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 16 / 26
∆ > 1
For ∆ > 1, the preceding argument does not work so well.
Instead we examine how a QOPT–knot K with OPTF interacts with aHeegaard torus T̂ of the lens space L(p, q).
If K is a torus knot (scc on T̂ ), then straightforward.Assume K is not a torus knot, then:
Put K into thin position with respect to T̂ .Show that K must be 1–bridge with respect to T̂ .Show that ∆ = 2 or ∆ = 3.Analyze resulting possible configurations.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 16 / 26
∆ > 1
DefinitionK ⊂ L(p, q) is in thin position if it minimizes this count:
Given a height function h : L(p, q)→ [−∞,+∞]
so that h−1(•) is a
{torus if • ∈ (−∞,+∞)
circle if • = ±∞for a • in each interval of regular values of h(K ),
add up |h−1(•) ∩ K |.
Thin position is a “tightened” bridge position.
With K in thin position, take Heegaard torus T̂ = h−1(0)where 0 is just above a minimum of K and just below a maximum of K .
t = |T̂ ∩ K |
T = T̂ −N(K )← delusions of being essential.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 17 / 26
∆ > 1
DefinitionK ⊂ L(p, q) is in thin position if it minimizes this count:
Given a height function h : L(p, q)→ [−∞,+∞]
so that h−1(•) is a
{torus if • ∈ (−∞,+∞)
circle if • = ±∞for a • in each interval of regular values of h(K ),
add up |h−1(•) ∩ K |.
Thin position is a “tightened” bridge position.
With K in thin position, take Heegaard torus T̂ = h−1(0)where 0 is just above a minimum of K and just below a maximum of K .
t = |T̂ ∩ K |
T = T̂ −N(K )← delusions of being essential.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 17 / 26
∆ > 1
DefinitionK ⊂ L(p, q) is in thin position if it minimizes this count:
Given a height function h : L(p, q)→ [−∞,+∞]
so that h−1(•) is a
{torus if • ∈ (−∞,+∞)
circle if • = ±∞for a • in each interval of regular values of h(K ),
add up |h−1(•) ∩ K |.
Thin position is a “tightened” bridge position.
With K in thin position, take Heegaard torus T̂ = h−1(0)where 0 is just above a minimum of K and just below a maximum of K .
t = |T̂ ∩ K |
T = T̂ −N(K )← delusions of being essential.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 17 / 26
∆ > 1
DefinitionK ⊂ L(p, q) is in thin position if it minimizes this count:
2
4
2
4
2
14
Given a height function h : L(p, q)→ [−∞,+∞]
so that h−1(•) is a
{torus if • ∈ (−∞,+∞)
circle if • = ±∞for a • in each interval of regular values of h(K ),
add up |h−1(•) ∩ K |.
Thin position is a “tightened” bridge position.
With K in thin position, take Heegaard torus T̂ = h−1(0)where 0 is just above a minimum of K and just below a maximum of K .
t = |T̂ ∩ K |
T = T̂ −N(K )← delusions of being essential.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 17 / 26
∆ > 1
DefinitionK ⊂ L(p, q) is in thin position if it minimizes this count:
2
4
2
4
2
14
Given a height function h : L(p, q)→ [−∞,+∞]
so that h−1(•) is a
{torus if • ∈ (−∞,+∞)
circle if • = ±∞for a • in each interval of regular values of h(K ),
add up |h−1(•) ∩ K |.
Thin position is a “tightened” bridge position.
With K in thin position, take Heegaard torus T̂ = h−1(0)where 0 is just above a minimum of K and just below a maximum of K .
t = |T̂ ∩ K |
T = T̂ −N(K )← delusions of being essential.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 17 / 26
∆ > 1
DefinitionK ⊂ L(p, q) is in thin position if it minimizes this count:
Given a height function h : L(p, q)→ [−∞,+∞]
so that h−1(•) is a
{torus if • ∈ (−∞,+∞)
circle if • = ±∞for a • in each interval of regular values of h(K ),
add up |h−1(•) ∩ K |.
Thin position is a “tightened” bridge position.
With K in thin position, take Heegaard torus T̂ = h−1(0)where 0 is just above a minimum of K and just below a maximum of K .
t = |T̂ ∩ K |
T = T̂ −N(K )← delusions of being essential.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 17 / 26
∆ > 1
DefinitionK ⊂ L(p, q) is in thin position if it minimizes this count:
Given a height function h : L(p, q)→ [−∞,+∞]
so that h−1(•) is a
{torus if • ∈ (−∞,+∞)
circle if • = ±∞for a • in each interval of regular values of h(K ),
add up |h−1(•) ∩ K |.
Thin position is a “tightened” bridge position.
With K in thin position, take Heegaard torus T̂ = h−1(0)where 0 is just above a minimum of K and just below a maximum of K .
t = |T̂ ∩ K |
T = T̂ −N(K )← delusions of being essential.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 17 / 26
∆ > 1
F = OPTand T = T̂ − N(K ) intersect nicely:
Form graphs GF on F and GT on T where:
Vertices are the punctures, andEdges are the arcs of F ∩ T .
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 18 / 26
∆ > 1
F = OPTand T = T̂ − N(K ) intersect nicely:
Form graphs GF on F and GT on T where:
Vertices are the punctures, andEdges are the arcs of F ∩ T .
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 18 / 26
∆ > 1
F = OPTand T = T̂ − N(K ) intersect nicely:
Form graphs GF on F and GT on T where:
Vertices are the punctures, andEdges are the arcs of F ∩ T .
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 18 / 26
∆ > 1
Thin position of K with choice of T̂=⇒ no monogons!
Vertices of GT have valence ∆=⇒ there are ∆t/2 Edges (on each GT and GF ).
Numbering the Vertices of GT in the order that K punctures T̂=⇒ numbering the endpoints of Edges of GF .
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 19 / 26
∆ > 1
Thin position of K with choice of T̂=⇒ no monogons!
Vertices of GT have valence ∆=⇒ there are ∆t/2 Edges (on each GT and GF ).
Numbering the Vertices of GT in the order that K punctures T̂=⇒ numbering the endpoints of Edges of GF .
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 19 / 26
∆ > 1
Thin position of K with choice of T̂=⇒ no monogons!
Vertices of GT have valence ∆=⇒ there are ∆t/2 Edges (on each GT and GF ).
Numbering the Vertices of GT in the order that K punctures T̂=⇒ numbering the endpoints of Edges of GF .
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 19 / 26
∆ > 1
A priori, the graph GF looks like one of the following:
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 20 / 26
∆ > 1
When t > 2, typically find long extended Scharlemann cycleswhich form long Mobius bands in the lens space M̂.
These “bind” too much of K=⇒ K is not in thin position.
With other contradictions =⇒ t = 2.(Also can eliminate simple closed curves of F ∩ T .)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 21 / 26
∆ > 1
When t > 2, typically find long extended Scharlemann cycleswhich form long Mobius bands in the lens space M̂.
These “bind” too much of K=⇒ K is not in thin position.
With other contradictions =⇒ t = 2.(Also can eliminate simple closed curves of F ∩ T .)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 21 / 26
∆ > 1
When t > 2, typically find long extended Scharlemann cycleswhich form long Mobius bands in the lens space M̂.
These “bind” too much of K=⇒ K is not in thin position.
With other contradictions =⇒ t = 2.(Also can eliminate simple closed curves of F ∩ T .)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 21 / 26
∆ > 1
When t > 2, typically find long extended Scharlemann cycleswhich form long Mobius bands in the lens space M̂.
These “bind” too much of K=⇒ K is not in thin position.
With other contradictions =⇒ t = 2.(Also can eliminate simple closed curves of F ∩ T .)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 21 / 26
∆ > 1
When t > 2, typically find long extended Scharlemann cycleswhich form long Mobius bands in the lens space M̂.
These “bind” too much of K=⇒ K is not in thin position.
With other contradictions =⇒ t = 2.(Also can eliminate simple closed curves of F ∩ T .)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 21 / 26
∆ > 1
When t > 2, typically find long extended Scharlemann cycleswhich form long Mobius bands in the lens space M̂.
These “bind” too much of K=⇒ K is not in thin position.
With other contradictions =⇒ t = 2.(Also can eliminate simple closed curves of F ∩ T .)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 21 / 26
∆ > 1
When t > 2, typically find long extended Scharlemann cycleswhich form long Mobius bands in the lens space M̂.
These “bind” too much of K=⇒ K is not in thin position.
With other contradictions =⇒ t = 2.(Also can eliminate simple closed curves of F ∩ T .)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 21 / 26
∆ > 1
When t > 2, typically find long extended Scharlemann cycleswhich form long Mobius bands in the lens space M̂.
These “bind” too much of K=⇒ K is not in thin position.
With other contradictions =⇒ t = 2.(Also can eliminate simple closed curves of F ∩ T .)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 21 / 26
∆ > 1
When t > 2, typically find long extended Scharlemann cycleswhich form long Mobius bands in the lens space M̂.
These “bind” too much of K=⇒ K is not in thin position.
With other contradictions =⇒ t = 2.(Also can eliminate simple closed curves of F ∩ T .)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 21 / 26
∆ > 1
When t > 2, typically find long extended Scharlemann cycleswhich form long Mobius bands in the lens space M̂.
These “bind” too much of K=⇒ K is not in thin position.
With other contradictions =⇒ t = 2.(Also can eliminate simple closed curves of F ∩ T .)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 21 / 26
∆ > 1
When t > 2, typically find long extended Scharlemann cycleswhich form long Mobius bands in the lens space M̂.
These “bind” too much of K=⇒ K is not in thin position.
With other contradictions =⇒ t = 2.(Also can eliminate simple closed curves of F ∩ T .)
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 21 / 26
∆ > 1
With t = 2, each face of GF is one of the following:
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 22 / 26
∆ > 1
With t = 2, each face of GF is one of the following:
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 22 / 26
∆ > 1
With t = 2, each face of GF is one of the following:
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 22 / 26
∆ > 1
With t = 2, each face of GF is one of the following:
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 22 / 26
∆ > 1
Studying how these pieces may fit together, we find two possibilities
∆ = 2GF GT
∆ = 3GF GT
In both situations, F is disjoint from a torus knot J
=⇒ meridians are not completely determined.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 23 / 26
∆ > 1
Studying how these pieces may fit together, we find two possibilities
∆ = 2GF GT
∆ = 3GF GT
In both situations, F is disjoint from a torus knot J
=⇒ meridians are not completely determined.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 23 / 26
∆ > 1
Studying how these pieces may fit together, we find two possibilities
∆ = 2GF GT
∆ = 3GF GT
In both situations, F is disjoint from a torus knot J
=⇒ meridians are not completely determined.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 23 / 26
∆ > 1
Studying how these pieces may fit together, we find two possibilities
∆ = 2GF GT
∆ = 3GF GT
In both situations, F is disjoint from a torus knot J
=⇒ meridians are not completely determined.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 23 / 26
∆ > 1
Get families of lens spaces with OPT–knots for ∆ = 2 and ∆ = 3.These torus knots J are OPT–knots, too.Exterior of K ∪ J is the exterior of the Whitehead link, W (·, ·).
TheoremUp to homeomorphism,K is a QOPT–knot in a lens space ⇐⇒ for k 6= 0 it is the core of oneof the following surgeries (except the −1 surgery)
The OPT–torus knots are the cores of the 6 + 1/k , −4 + 1/k , and−3 + 1/k surgeries.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 24 / 26
∆ > 1
Get families of lens spaces with OPT–knots for ∆ = 2 and ∆ = 3.These torus knots J are OPT–knots, too.Exterior of K ∪ J is the exterior of the Whitehead link, W (·, ·).
TheoremUp to homeomorphism,K is a QOPT–knot in a lens space ⇐⇒ for k 6= 0 it is the core of oneof the following surgeries (except the −1 surgery)
The OPT–torus knots are the cores of the 6 + 1/k , −4 + 1/k , and−3 + 1/k surgeries.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 24 / 26
∆ > 1
Get families of lens spaces with OPT–knots for ∆ = 2 and ∆ = 3.These torus knots J are OPT–knots, too.Exterior of K ∪ J is the exterior of the Whitehead link, W (·, ·).
TheoremUp to homeomorphism,K is a QOPT–knot in a lens space ⇐⇒ for k 6= 0 it is the core of oneof the following surgeries (except the −1 surgery)
The OPT–torus knots are the cores of the 6 + 1/k , −4 + 1/k , and−3 + 1/k surgeries.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 24 / 26
∆ > 1
Get families of lens spaces with OPT–knots for ∆ = 2 and ∆ = 3.These torus knots J are OPT–knots, too.Exterior of K ∪ J is the exterior of the Whitehead link, W (·, ·).
TheoremUp to homeomorphism,K is a QOPT–knot in a lens space ⇐⇒ for k 6= 0 it is the core of oneof the following surgeries (except the −1 surgery)
The OPT–torus knots are the cores of the 6 + 1/k , −4 + 1/k , and−3 + 1/k surgeries.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 24 / 26
∆ > 1
Get families of lens spaces with OPT–knots for ∆ = 2 and ∆ = 3.These torus knots J are OPT–knots, too.Exterior of K ∪ J is the exterior of the Whitehead link, W (·, ·).
TheoremUp to homeomorphism,K is a QOPT–knot in a lens space ⇐⇒ for k 6= 0 it is the core of oneof the following surgeries (except the −1 surgery)
−2
−4+1/k
−3
−3+1/k
The OPT–torus knots are the cores of the 6 + 1/k , −4 + 1/k , and−3 + 1/k surgeries.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 24 / 26
∆ > 1
Get families of lens spaces with OPT–knots for ∆ = 2 and ∆ = 3.These torus knots J are OPT–knots, too.Exterior of K ∪ J is the exterior of the Whitehead link, W (·, ·).
TheoremUp to homeomorphism,K is a QOPT–knot in a lens space ⇐⇒ for k 6= 0 it is the core of oneof the following surgeries (except the −1 surgery)
−1
−6+1/k
−2
−4+1/k
−3
−3+1/k
The OPT–torus knots are the cores of the 6 + 1/k , −4 + 1/k , and−3 + 1/k surgeries.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 24 / 26
∆ > 1
FactW (γ, ·) is a OPT–bundle ⇐⇒ γ ∈ Z
TheoremIf K is a QOPT–fibered knot with monodromy φ in a lens space,then for k 6= 0 we have
K ⊂
L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)
L(12, 5)L(10, 3)
and φ ∼=
τxτy
τ2x τy
τ3x τy
τ4x τy
τ5x τy
.
← Trefoil← torus knot← torus knot← Klein bottle← Fig. Eight Sis.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 25 / 26
∆ > 1
FactW (γ, ·) is a OPT–bundle ⇐⇒ γ ∈ Z
−1
−6+1/k
−2
−4+1/k
−3
−3+1/k
TheoremIf K is a QOPT–fibered knot with monodromy φ in a lens space,then for k 6= 0 we have
K ⊂
L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)
L(12, 5)L(10, 3)
and φ ∼=
τxτy
τ2x τy
τ3x τy
τ4x τy
τ5x τy
.
← Trefoil← torus knot← torus knot← Klein bottle← Fig. Eight Sis.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 25 / 26
∆ > 1
FactW (γ, ·) is a OPT–bundle ⇐⇒ γ ∈ Z
−1
−6+1/k
−2
−4+1/k
−3
−3+1/k
TheoremIf K is a QOPT–fibered knot with monodromy φ in a lens space,then for k 6= 0 we have
K ⊂
L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)
L(12, 5)L(10, 3)
and φ ∼=
τxτy
τ2x τy
τ3x τy
τ4x τy
τ5x τy
.
← Trefoil← torus knot← torus knot← Klein bottle← Fig. Eight Sis.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 25 / 26
∆ > 1
FactW (γ, ·) is a OPT–bundle ⇐⇒ γ ∈ Z
−1
−6+1/k
−2
−4+1/k
−3
−3+1/k
TheoremIf K is a QOPT–fibered knot with monodromy φ in a lens space,then for k 6= 0 we have
K ⊂
L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)
L(12, 5)L(10, 3)
and φ ∼=
τxτy
τ2x τy
τ3x τy
τ4x τy
τ5x τy
.
← Trefoil← torus knot← torus knot← Klein bottle← Fig. Eight Sis.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 25 / 26
∆ > 1
FactW (γ, ·) is a OPT–bundle ⇐⇒ γ ∈ Z
−1
−6+1/k
−2
−4+1/k
−3
−3+1/k
TheoremIf K is a QOPT–fibered knot with monodromy φ in a lens space,then for k 6= 0 we have
K ⊂
L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)
L(12, 5)L(10, 3)
and φ ∼=
τxτy
τ2x τy
τ3x τy
τ4x τy
τ5x τy
.
← Trefoil← torus knot← torus knot← Klein bottle← Fig. Eight Sis.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 25 / 26
∆ > 1
FactW (γ, ·) is a OPT–bundle ⇐⇒ γ ∈ Z
−1
−6+1/k
−2
−4+1/k
−3
−3+1/k
TheoremIf K is a QOPT–fibered knot with monodromy φ in a lens space,then for k 6= 0 we have
K ⊂
L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)
L(12, 5)L(10, 3)
and φ ∼=
τxτy
τ2x τy
τ3x τy
τ4x τy
τ5x τy
.
← Trefoil← torus knot← torus knot← Klein bottle← Fig. Eight Sis.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 25 / 26
∆ > 1
FactW (γ, ·) is a OPT–bundle ⇐⇒ γ ∈ Z
−1
−6+1/k
−2
−4+1/k
−3
−3+1/k
TheoremIf K is a QOPT–fibered knot with monodromy φ in a lens space,then for k 6= 0 we have
K ⊂
L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)
L(12, 5)L(10, 3)
and φ ∼=
τxτy
τ2x τy
τ3x τy
τ4x τy
τ5x τy
.
← Trefoil← torus knot← torus knot← Klein bottle← Fig. Eight Sis.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 25 / 26
∆ > 1
FactW (γ, ·) is a OPT–bundle ⇐⇒ γ ∈ Z
−1
−6+1/k
−2
−4+1/k
−3
−3+1/k
TheoremIf K is a QOPT–fibered knot with monodromy φ in a lens space,then for k 6= 0 we have
K ⊂
L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)
L(12, 5)L(10, 3)
and φ ∼=
τxτy
τ2x τy
τ3x τy
τ4x τy
τ5x τy
.
← Trefoil← torus knot← torus knot← Klein bottle← Fig. Eight Sis.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 25 / 26
∆ > 1
FactW (γ, ·) is a OPT–bundle ⇐⇒ γ ∈ Z
−1
−6+1/k
−2
−4+1/k
−3
−3+1/k
TheoremIf K is a QOPT–fibered knot with monodromy φ in a lens space,then for k 6= 0 we have
K ⊂
L(6k − 1, 2k − 1)L(8k − 2, 4k + 1)L(9k − 3, 3k − 2)
L(12, 5)L(10, 3)
and φ ∼=
τxτy
τ2x τy
τ3x τy
τ4x τy
τ5x τy
.
← Trefoil← torus knot← torus knot← Klein bottle← Fig. Eight Sis.
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 25 / 26
Lens space fillings of OPT–bundles: ∆ = 1 and ∆ > 1
TheoremM is an OPT–bundle admitting a lens space filling of distance ∆
m
M ∼= F × I/φ with φ = τnx τ2
y τmx τ−1
y for some m, n ∈ Z.
Moreover, ∆ = 1 with the following exceptions:
φ = τxτy and ∆ = | − 6k + 1| for k ∈ Z
φ = τ2x τy and ∆ = | − 4k + 1| for k ∈ Z
φ = τ3x τy and ∆ = | − 3k + 1| for k ∈ Z
φ = τ4x τy and ∆ = 1, 3
φ = τ5x τy and ∆ = 1, 2
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 26 / 26
Lens space fillings of OPT–bundles: ∆ = 1 and ∆ > 1
TheoremM is an OPT–bundle admitting a lens space filling of distance ∆
m
M ∼= F × I/φ with φ = τnx τ2
y τmx τ−1
y for some m, n ∈ Z.
Moreover, ∆ = 1 with the following exceptions:
φ = τxτy and ∆ = | − 6k + 1| for k ∈ Z
φ = τ2x τy and ∆ = | − 4k + 1| for k ∈ Z
φ = τ3x τy and ∆ = | − 3k + 1| for k ∈ Z
φ = τ4x τy and ∆ = 1, 3
φ = τ5x τy and ∆ = 1, 2
Kenneth L. Baker (GaTech) Lens space fillings of OPT–bundles AMS-SMM ’07 26 / 26