surfaces in seifert fibered spaces

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Surfaces in Seifert fibered spaces Jennifer Schultens University of California, Davis June 26, 2019 Jennifer Schultens Surfaces in Seifert fibered spaces

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Page 1: Surfaces in Seifert fibered spaces

Surfaces in Seifert fibered spaces

Jennifer Schultens

University of California, Davis

June 26, 2019

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 2: Surfaces in Seifert fibered spaces

General comments

Low-dimensional topology concerns the topology of manifolds in 1,2, 3, and 4 dimensions.

We also study submanifolds. E.g., 1-dimensional knots in3-dimensional space.

I will discuss the totality of surfaces in a particular class of3-manifolds.

Rather than giving you a formal definition of manifold, in particular3-manifold, I will describe some of my favorite examples.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 3: Surfaces in Seifert fibered spaces

General comments

Low-dimensional topology concerns the topology of manifolds in 1,2, 3, and 4 dimensions.

We also study submanifolds. E.g., 1-dimensional knots in3-dimensional space.

I will discuss the totality of surfaces in a particular class of3-manifolds.

Rather than giving you a formal definition of manifold, in particular3-manifold, I will describe some of my favorite examples.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 4: Surfaces in Seifert fibered spaces

General comments

Low-dimensional topology concerns the topology of manifolds in 1,2, 3, and 4 dimensions.

We also study submanifolds. E.g., 1-dimensional knots in3-dimensional space.

I will discuss the totality of surfaces in a particular class of3-manifolds.

Rather than giving you a formal definition of manifold, in particular3-manifold, I will describe some of my favorite examples.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 5: Surfaces in Seifert fibered spaces

General comments

Low-dimensional topology concerns the topology of manifolds in 1,2, 3, and 4 dimensions.

We also study submanifolds. E.g., 1-dimensional knots in3-dimensional space.

I will discuss the totality of surfaces in a particular class of3-manifolds.

Rather than giving you a formal definition of manifold, in particular3-manifold, I will describe some of my favorite examples.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 6: Surfaces in Seifert fibered spaces

Examples of manifolds: Not knot

Definition

Let K be a knot. The complement of K is

C (K ) = S3 − η(K )

where η(K ) is a regular neighborhood of K .

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 7: Surfaces in Seifert fibered spaces

Not knot

Figure: The complement of the unknot

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 8: Surfaces in Seifert fibered spaces

Not knot

Figure: The complement of a knot

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 9: Surfaces in Seifert fibered spaces

Torus knots

Figure: T(2, 3), also known as the trefoill

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 10: Surfaces in Seifert fibered spaces

Satellite knots

I-’

4KC sot’? kftco V

(Jattant

i2

KcV•

Figure: Satellite knots

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 11: Surfaces in Seifert fibered spaces

Thurston’s theorem

Theorem

(Trichotomy for knots) Let K be a knot in S3. Then exactly one ofthe following holds:

K is a torus knot;

K is a satellite knot;

K is hyperbolic.

A proof can be found in M. Kapovich’s “Hyperbolic manifolds anddiscrete groups”.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 12: Surfaces in Seifert fibered spaces

Thurston’s theorem

Theorem

(Trichotomy for knots) Let K be a knot in S3. Then exactly one ofthe following holds:

K is a torus knot;

K is a satellite knot;

K is hyperbolic.

A proof can be found in M. Kapovich’s “Hyperbolic manifolds anddiscrete groups”.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 13: Surfaces in Seifert fibered spaces

Examples of manifolds: 2-fold branched cover

2-fold branched cover of a knot: The 2-fold branched cover of aknot has two points for every point in the knot complement andone point for every point on the knot.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 14: Surfaces in Seifert fibered spaces

Examples of manifolds: 2-fold branched cover

Figure: A Seifert surface

Theorem

(Seifert) Every knot admits a Seifert surface.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 15: Surfaces in Seifert fibered spaces

Examples of manifolds: Seifert fibered spaces

Seifert fibered spaces are a family of 3-dimensional spaces(3-manifolds) whose members are classified by a finite set ofinvariants. The structure of these manifolds allows us to concretelydescribe essential surfaces embedded in them.

We are interested in the collection of all surfaces in a given Seifertfibered space. However, we must restrict our attention to essentialsurfaces in Seifert fibered spaces. The Kakimizu complex of a knotprovides an example of how to encode surfaces. We discus theanalogous concept for Seifert fibered spaces.

We will always assume that curves in surfaces are simple and thatsurfaces in 3-manifolds are properly embedded.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 16: Surfaces in Seifert fibered spaces

Fibered solid tori

identify after twist

Figure: A fibered solid torus

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 17: Surfaces in Seifert fibered spaces

Fibered solid tori

Let ν ∈ Z and µ ∈ N be such that g .c .d .(ν, µ) = 1.

The fibered solid torus V (ν, µ) is obtained from the solid cylinderD2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2

about its center point x by 2πνµ .

Intervals of the form y × [0, 1] for y ∈ D2 match up to form simpleclosed curves called fibers.

All fibers except the one determined by x × I traverse the torus µtimes. The curve x × S1 is called the central fiber.

If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber.

The other fibers are called regular fibers. If µ = 1 or ν = 0, thenthe central fiber is also called a regular fiber.

Two fibered solid tori are considered equivalent if they arehomeomorphic via a fiber preserving homeomorphism.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 18: Surfaces in Seifert fibered spaces

Fibered solid tori

Let ν ∈ Z and µ ∈ N be such that g .c .d .(ν, µ) = 1.

The fibered solid torus V (ν, µ) is obtained from the solid cylinderD2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2

about its center point x by 2πνµ .

Intervals of the form y × [0, 1] for y ∈ D2 match up to form simpleclosed curves called fibers.

All fibers except the one determined by x × I traverse the torus µtimes. The curve x × S1 is called the central fiber.

If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber.

The other fibers are called regular fibers. If µ = 1 or ν = 0, thenthe central fiber is also called a regular fiber.

Two fibered solid tori are considered equivalent if they arehomeomorphic via a fiber preserving homeomorphism.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 19: Surfaces in Seifert fibered spaces

Fibered solid tori

Let ν ∈ Z and µ ∈ N be such that g .c .d .(ν, µ) = 1.

The fibered solid torus V (ν, µ) is obtained from the solid cylinderD2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2

about its center point x by 2πνµ .

Intervals of the form y × [0, 1] for y ∈ D2 match up to form simpleclosed curves called fibers.

All fibers except the one determined by x × I traverse the torus µtimes. The curve x × S1 is called the central fiber.

If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber.

The other fibers are called regular fibers. If µ = 1 or ν = 0, thenthe central fiber is also called a regular fiber.

Two fibered solid tori are considered equivalent if they arehomeomorphic via a fiber preserving homeomorphism.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 20: Surfaces in Seifert fibered spaces

Fibered solid tori

Let ν ∈ Z and µ ∈ N be such that g .c .d .(ν, µ) = 1.

The fibered solid torus V (ν, µ) is obtained from the solid cylinderD2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2

about its center point x by 2πνµ .

Intervals of the form y × [0, 1] for y ∈ D2 match up to form simpleclosed curves called fibers.

All fibers except the one determined by x × I traverse the torus µtimes. The curve x × S1 is called the central fiber.

If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber.

The other fibers are called regular fibers. If µ = 1 or ν = 0, thenthe central fiber is also called a regular fiber.

Two fibered solid tori are considered equivalent if they arehomeomorphic via a fiber preserving homeomorphism.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 21: Surfaces in Seifert fibered spaces

Fibered solid tori

Let ν ∈ Z and µ ∈ N be such that g .c .d .(ν, µ) = 1.

The fibered solid torus V (ν, µ) is obtained from the solid cylinderD2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2

about its center point x by 2πνµ .

Intervals of the form y × [0, 1] for y ∈ D2 match up to form simpleclosed curves called fibers.

All fibers except the one determined by x × I traverse the torus µtimes. The curve x × S1 is called the central fiber.

If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber.

The other fibers are called regular fibers. If µ = 1 or ν = 0, thenthe central fiber is also called a regular fiber.

Two fibered solid tori are considered equivalent if they arehomeomorphic via a fiber preserving homeomorphism.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 22: Surfaces in Seifert fibered spaces

Fibered solid tori

Let ν ∈ Z and µ ∈ N be such that g .c .d .(ν, µ) = 1.

The fibered solid torus V (ν, µ) is obtained from the solid cylinderD2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2

about its center point x by 2πνµ .

Intervals of the form y × [0, 1] for y ∈ D2 match up to form simpleclosed curves called fibers.

All fibers except the one determined by x × I traverse the torus µtimes. The curve x × S1 is called the central fiber.

If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber.

The other fibers are called regular fibers. If µ = 1 or ν = 0, thenthe central fiber is also called a regular fiber.

Two fibered solid tori are considered equivalent if they arehomeomorphic via a fiber preserving homeomorphism.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 23: Surfaces in Seifert fibered spaces

Fibered solid tori

Let ν ∈ Z and µ ∈ N be such that g .c .d .(ν, µ) = 1.

The fibered solid torus V (ν, µ) is obtained from the solid cylinderD2 × [0, 1] by gluing D2 × {0} to D2 × {1} after rotating D2

about its center point x by 2πνµ .

Intervals of the form y × [0, 1] for y ∈ D2 match up to form simpleclosed curves called fibers.

All fibers except the one determined by x × I traverse the torus µtimes. The curve x × S1 is called the central fiber.

If µ > 1 and ν 6= 0, then the central fiber is called a singular fiber.

The other fibers are called regular fibers. If µ = 1 or ν = 0, thenthe central fiber is also called a regular fiber.

Two fibered solid tori are considered equivalent if they arehomeomorphic via a fiber preserving homeomorphism.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 24: Surfaces in Seifert fibered spaces

Fibered solid tori

Every fiber in a fibered solid torus except the singular onerepresents the element µ ∈ π1(V (ν, µ)) = Z which is generated bythe class of the central fiber.

Hence µ is an invariant of V (ν, µ) and ν is an invariant up to signand mod µ.

In other words, V (ν, µ) is isomorphic to V (ν ′, µ′) via a fiberpreserving homeomorphism if and only if µ′ = µ andν ′ = ±ν ′(modµ).

If we keep track of orientations we can normalize the invariant ν sothat 0 ≤ ν < µ.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 25: Surfaces in Seifert fibered spaces

Fibered solid tori

Every fiber in a fibered solid torus except the singular onerepresents the element µ ∈ π1(V (ν, µ)) = Z which is generated bythe class of the central fiber.

Hence µ is an invariant of V (ν, µ) and ν is an invariant up to signand mod µ.

In other words, V (ν, µ) is isomorphic to V (ν ′, µ′) via a fiberpreserving homeomorphism if and only if µ′ = µ andν ′ = ±ν ′(modµ).

If we keep track of orientations we can normalize the invariant ν sothat 0 ≤ ν < µ.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 26: Surfaces in Seifert fibered spaces

Fibered solid tori

Every fiber in a fibered solid torus except the singular onerepresents the element µ ∈ π1(V (ν, µ)) = Z which is generated bythe class of the central fiber.

Hence µ is an invariant of V (ν, µ) and ν is an invariant up to signand mod µ.

In other words, V (ν, µ) is isomorphic to V (ν ′, µ′) via a fiberpreserving homeomorphism if and only if µ′ = µ andν ′ = ±ν ′(modµ).

If we keep track of orientations we can normalize the invariant ν sothat 0 ≤ ν < µ.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 27: Surfaces in Seifert fibered spaces

Fibered solid tori

Every fiber in a fibered solid torus except the singular onerepresents the element µ ∈ π1(V (ν, µ)) = Z which is generated bythe class of the central fiber.

Hence µ is an invariant of V (ν, µ) and ν is an invariant up to signand mod µ.

In other words, V (ν, µ) is isomorphic to V (ν ′, µ′) via a fiberpreserving homeomorphism if and only if µ′ = µ andν ′ = ±ν ′(modµ).

If we keep track of orientations we can normalize the invariant ν sothat 0 ≤ ν < µ.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 28: Surfaces in Seifert fibered spaces

Seifert fibered spaces

For simplicity, we consider only orientable manifolds.

Definition

A Seifert Fibered Space is a compact connected 3-manifold M thatis a union of disjoint circles called fibers such that each fiber has aneighborhood that is homeomorphic to a fibered solid torus.

Definition

Fibers are called regular (or singular) if they are regular (orsingular) fibers in the fibered solid torus containing them.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 29: Surfaces in Seifert fibered spaces

Seifert fibered spaces

For simplicity, we consider only orientable manifolds.

Definition

A Seifert Fibered Space is a compact connected 3-manifold M thatis a union of disjoint circles called fibers such that each fiber has aneighborhood that is homeomorphic to a fibered solid torus.

Definition

Fibers are called regular (or singular) if they are regular (orsingular) fibers in the fibered solid torus containing them.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 30: Surfaces in Seifert fibered spaces

Seifert fibered spaces

Remark 1: Fibers of a Seifert fibered space are either regular orsingular. (Not both.) Moreover, a fiber of a Seifert fibered spaceuniquely determines normalized ν, µ.

Remark 2: Because it is compact, a Seifert fibered space is coveredby finitely many fibered solid tori.

Remark 3: A Seifert fibered space will have only a finite number ofsingular fibers.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 31: Surfaces in Seifert fibered spaces

Seifert fibered spaces

Remark 1: Fibers of a Seifert fibered space are either regular orsingular. (Not both.) Moreover, a fiber of a Seifert fibered spaceuniquely determines normalized ν, µ.

Remark 2: Because it is compact, a Seifert fibered space is coveredby finitely many fibered solid tori.

Remark 3: A Seifert fibered space will have only a finite number ofsingular fibers.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 32: Surfaces in Seifert fibered spaces

Seifert fibered spaces

Remark 1: Fibers of a Seifert fibered space are either regular orsingular. (Not both.) Moreover, a fiber of a Seifert fibered spaceuniquely determines normalized ν, µ.

Remark 2: Because it is compact, a Seifert fibered space is coveredby finitely many fibered solid tori.

Remark 3: A Seifert fibered space will have only a finite number ofsingular fibers.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 33: Surfaces in Seifert fibered spaces

Seifert fibered spaces

Given a Seifert fibered space, we identify each fiber to a point toobtain the base orbifold, a compact surface with a finite number ofmarked points.

(We will assume that this surface is orientable.)

The marked points are called singular points.

Remark: A closed oriented Seifert fibered space M is completelydetermined by a set of invariants called a signature:

{g , b, e | (α1, β1), . . . , (αr , βr )}

(More on the Euler number e in a moment.)

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 34: Surfaces in Seifert fibered spaces

Seifert fibered spaces

Given a Seifert fibered space, we identify each fiber to a point toobtain the base orbifold, a compact surface with a finite number ofmarked points.

(We will assume that this surface is orientable.)

The marked points are called singular points.

Remark: A closed oriented Seifert fibered space M is completelydetermined by a set of invariants called a signature:

{g , b, e | (α1, β1), . . . , (αr , βr )}

(More on the Euler number e in a moment.)

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 35: Surfaces in Seifert fibered spaces

Seifert fibered spaces

Given a Seifert fibered space, we identify each fiber to a point toobtain the base orbifold, a compact surface with a finite number ofmarked points.

(We will assume that this surface is orientable.)

The marked points are called singular points.

Remark: A closed oriented Seifert fibered space M is completelydetermined by a set of invariants called a signature:

{g , b, e | (α1, β1), . . . , (αr , βr )}

(More on the Euler number e in a moment.)

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 36: Surfaces in Seifert fibered spaces

Seifert fibered spaces

Given a Seifert fibered space, we identify each fiber to a point toobtain the base orbifold, a compact surface with a finite number ofmarked points.

(We will assume that this surface is orientable.)

The marked points are called singular points.

Remark: A closed oriented Seifert fibered space M is completelydetermined by a set of invariants called a signature:

{g , b, e | (α1, β1), . . . , (αr , βr )}

(More on the Euler number e in a moment.)

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 37: Surfaces in Seifert fibered spaces

Seifert fibered spaces

Figure: A base orbifold for a Seifert fibered space

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 38: Surfaces in Seifert fibered spaces

Seifert fibered spaces

Figure: A base orbifold for a Seifert fibered space

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 39: Surfaces in Seifert fibered spaces

Compressible/incompressible surfaces

Definition

The simple closed curve α in a surface F is inessential if it boundsa disk. If it is not inessential, then it is essential.

Definition

The surface F in the 3-manifold M is compressible if there is anessential simple closed curve in F that bounds a disk in M. If F isnot compressible, then it is incompressible.

Jennifer Schultens Surfaces in Seifert fibered spaces

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Compressible/incompressible surfaces

D

Figure: A compressible surface

Jennifer Schultens Surfaces in Seifert fibered spaces

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Incompressible surfaces in Seifert fibered spaces

Definition

If the surface F is everywhere transverse to the fibers of the Seifertfibered space M, then F is said to be horizontal.

1

x [0, 1] identify

Figure: A surface bundle over the circle

Jennifer Schultens Surfaces in Seifert fibered spaces

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Incompressible surfaces in Seifert fibered spaces

Remark 1: A horizontal surface in a Seifert fibered space isnecessarily incompressible.

Remark 2: A Seifert fibered space admits horizontal surfaces if andonly if its Euler number is 0.

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 43: Surfaces in Seifert fibered spaces

Incompressible surfaces in Seifert fibered spaces

Definition

If every fiber of the Seifert fibered space M that meets the surfaceF is entirely contained in F , then F is said to be vertical.

Figure: A vertical surface in a Seifert fibered space

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 44: Surfaces in Seifert fibered spaces

Incompressible surfaces in Seifert fibered spaces

CAUTION:

Figure: Not a vertical surface in an orientable Seifert fibered space

Jennifer Schultens Surfaces in Seifert fibered spaces

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Incompressible surfaces in Seifert fibered spaces

πidentify after twist through

Figure: Only a vertical Mobius band in a fibered solid Klein bottle wouldproject to an arc

Jennifer Schultens Surfaces in Seifert fibered spaces

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Incompressible surfaces in Seifert fibered spaces

Remark: A vertical surface in a Seifert fibered space can becompressible or incompressible. If it is compressible, then it boundsa fibered solid torus. (Dehn’s lemma + Loop theorem)

Jennifer Schultens Surfaces in Seifert fibered spaces

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A theorem of Jaco

Theorem

(Jaco) Let M be an orientable Seifert fibered space with orientablebase orbifold. If F is a connected, two-sided, incompressiblesurface in M, then one of the following alternatives holds:

(i) F is a disk or an annulus and F is parallel into ∂M;

(ii) F does not separate M and F is a fiber in a fiberation of M asa surface bundle over the circle (in particular, F is horizontal);

(iii) F is an annulus or a torus and, after isotopy, F consists offibers, in some Seifert fiberation of M.

Jennifer Schultens Surfaces in Seifert fibered spaces

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The Kakimizu complex

Introduction to the Kakimizu complex

Jennifer Schultens Surfaces in Seifert fibered spaces

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Seifert surfaces

The Kakimizu complex evolved from the study of spanningsurfaces for knots.

Definition

Given an knot K in S3, a compact connected orientable surfacethat represents a generator of H2(S3,K ) is called a Seifert surface.More concretely, a Seifert surface is an embedded orientablesurface S such that ∂S = K .

Jennifer Schultens Surfaces in Seifert fibered spaces

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Seifert Surfaces

Figure: A Seifert surface

Theorem

(Seifert) Every knot admits a Seifert surface.

Jennifer Schultens Surfaces in Seifert fibered spaces

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Seifert Surfaces

Figure: Two knots

Theorem

(Eisner) Many knots K in S3 admit non-isotopic Seifert surfaces.

Jennifer Schultens Surfaces in Seifert fibered spaces

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Seifert Surfaces

Figure: Connect sum of knots with swallow-follow torus

Jennifer Schultens Surfaces in Seifert fibered spaces

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Seifert Surfaces

Figure: Schematic for spinning Seifert surface around swallow-follow torus

Jennifer Schultens Surfaces in Seifert fibered spaces

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Kakimizu complex

Definition

The vertices of the Kakimizu complex Kak(K ) of a knot K in S3

are given by the isotopy classes of minimal genus Seifert surfacesfor K .

The n-simplices of the Kakimizu complex of K , for n > 1, are givenby n-tuples of vertices that admit pairwise disjoint representatives.

Jennifer Schultens Surfaces in Seifert fibered spaces

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Examples of Kakimizu complexes

Example I: Fibered knots have trivial Kakimizu complexes.

Example II: Hyperbolic knots have finite Kakimizu complexes.

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Examples of Kakimizu complexes

Example I: Fibered knots have trivial Kakimizu complexes.

Example II: Hyperbolic knots have finite Kakimizu complexes.

Jennifer Schultens Surfaces in Seifert fibered spaces

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Topology of the Kakimizu complex

Theorem

(Scharlemann-Thompson, Kakimizu) The Kakimizu complex isconnected.

Theorem

(S) The Kakimizu complex of a knot K is a flag complex.

Jennifer Schultens Surfaces in Seifert fibered spaces

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Topology of the Kakimizu complex

Theorem

(Banks) The Kakimizu complex of a knot is not necessarily locallyfinite.

Theorem

(Banks) Approximately: The Kakimizu complex of a sum of twoknots is the product of their Kakimizu complexes with Z.

Jennifer Schultens Surfaces in Seifert fibered spaces

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Geometry of the Kakimizu complex

Theorem

(Przytycki-S) The Kakimizu complex of a knot is contractible.

Theorem

(Johnson-Pelayo-Wilson) The Kakimizu complex of a knot isquasi-Euclidean.

Jennifer Schultens Surfaces in Seifert fibered spaces

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Just for fun: Find a Seifert surface

Figure: Whitehead double of the figure eight knot

Jennifer Schultens Surfaces in Seifert fibered spaces

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What’s special about Seifert surfaces of knots?

Figure: A Seifert surface

Jennifer Schultens Surfaces in Seifert fibered spaces

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What’s special about Seifert surfaces of knots?

Key Property: They represent the generator ofH2(S3,K ) = H2(S3 − η(K ), ∂S3 − η(K )).

Fact: This group is dual toH1(S3 − η(K )) = Hom(H1(S3 − η(K )) = Hom(Z) = Z

Consequence: There is a canonical infinite cyclic cover of(S3,K ) in which we can compare lifts of distinct Seifertsurfaces.

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What’s special about Seifert surfaces of knots?

Key Property: They represent the generator ofH2(S3,K ) = H2(S3 − η(K ), ∂S3 − η(K )).

Fact: This group is dual toH1(S3 − η(K )) = Hom(H1(S3 − η(K )) = Hom(Z) = Z

Consequence: There is a canonical infinite cyclic cover of(S3,K ) in which we can compare lifts of distinct Seifertsurfaces.

Jennifer Schultens Surfaces in Seifert fibered spaces

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What’s special about Seifert surfaces of knots?

Key Property: They represent the generator ofH2(S3,K ) = H2(S3 − η(K ), ∂S3 − η(K )).

Fact: This group is dual toH1(S3 − η(K )) = Hom(H1(S3 − η(K )) = Hom(Z) = Z

Consequence: There is a canonical infinite cyclic cover of(S3,K ) in which we can compare lifts of distinct Seifertsurfaces.

Jennifer Schultens Surfaces in Seifert fibered spaces

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The covering space associated with α

S’0

m

m−2

S

m−1S

S

Figure: No edge of a Kakimizu complex

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What’s special about Seifert surfaces?

Generalization of Key Property: In an orientable 3-manifoldM3, choose a primitive element α of H2(M3, ∂M3). Or, moregenerally, choose a primitive element α of Hn−1(Mn, ∂Mn).

Jennifer Schultens Surfaces in Seifert fibered spaces

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Kakimizu complex of a 3-manifold

Let α be a primitive element of H2(M, ∂M,Z).

Definition

We define the Kakimizu complex of M with respect to α, denotedKak(M, α): Vertices are given by weighted multi-surfaces thatrepresent α.

The multi-surface is required to be Thurston norm minimizing, tohave connected complement, and is considered up to isotopy.

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Kakimizu complex of a 3-manifold

Definition

The vertices v , v ′ span an edge e = (v , v ′) if and only ifrepresentatives of v , v ′ can be chosen so that the complement of alift of one respresentative to the covering space associated with αintersects exactly two lifts of the complement of the otherrepresentative.

(This condition implies that the representatives are disjoint, butnot vice versa.)

Kak(M, α) is the flag complex with the vertices and edgesdescribed above. (I.e., add simplicies whenever possible.)

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The topology and geometry of Kakimizu complexes of 2-and 3-manifolds

Theorem

(Scharlemann-Thompson, Kakimizu, Przytycki-S, S) EveryKakimizu complex of a 2- or 3-manifold is connected.

Theorem

(Przytycki-S, S) Every Kakimizu complex of a 2- or 3-manifold iscontractible.

Theorem

(S) A Kakimizu complex of a 2- or 3-manifold need not be quasiEuclidean.

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The Kakimizu complexes of Seifert fibered spaces

The Kakimizu complexes of Seifert fibered spaces

Jennifer Schultens Surfaces in Seifert fibered spaces

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Horizontal Kakimizu complexes of Seifert fibered spaces

Theorem

Let M be an orientable Seifert fibered space with orientable baseorbifold. Let α ∈ H2(M, ∂M,Z) be a primitive relative secondhomology class that is represented by a horizontal surface. ThenKak(M, α) is trivial, i.e., consists of a single vertex.

This result also follows from work of Jaco.

Question: What if α is represented by a vertical surface?

Jennifer Schultens Surfaces in Seifert fibered spaces

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Vertical Kakimizu complexes of Seifert fibered spaces

Theorem

(S) Let M be a Seifert fibered space with orientable base space.Let α ∈ H2(M, ∂M,Z) be a homology class that is represented bya vertical surface. Then Kak(M, α) is isomorphic to thecorresponding Kakimizu complex of the surface obtained from thebase orbifold of M by removing neighborhoods of the singularpoints.

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Vertical Kakimizu complexes of Seifert fibered spaces

−ξ

~

b~

A

b’~

ξ

Figure: A lune

Jennifer Schultens Surfaces in Seifert fibered spaces

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The surface complex of a Seifert fibered space

The surface complex of a Seifert fibered space

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 75: Surfaces in Seifert fibered spaces

Finegold’s torus complex

In her dissertation, written under the direction of Daryl Cooper in2010, Brie Finegold studied torus complexes in all dimensions froman algebraic point of view. In dimension 2, where vertices aresimple closed curves in the 2-torus, Finegold’s torus complexcoincides with the curve complex of the torus, i.e., the Farey graph.

In dimension 3, where the vertices are isotopy classes of 2-toriembedded in the 3-torus, there are no disjoint non isotopicessential surfaces, and edges are defined as pairs of vertices withrepresentatives meeting in a single simple closed curve.

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Finegold’s torus complex

Theorem

(Finegold) The torus complex in dimension 3 is connected.

Theorem

(Finegold) The torus complex in dimension 3 is simply-connected.

Theorem

(Finegold) The torus complex in dimension 3 has diameter 2.

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The surface complex

Let M be a compact orientable 3-manifold. We define a sequenceof complexes {Si (M)}, and the surface complex, S(M), of M asfollows:

Vertices in {Si (M)} and S(M) correspond to isotopy classesof compact connected orientable essential (incompressible,boundary incompressible and not boundary parallel) surfaces(properly embedded) in M.

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The surface complex

A pair of distinct vertices (v1, v2) spans an edge in S0(M) ifand only if v1 and v2 admit disjoint representatives.Inductively, we construct a sequence of complexes, {Si (M)},whose vertices coincide, for all i , with those of S0(M). GivenSi (M), the pair of vertices {v1, v2} spans an edge in Si+1(M)if and only if v1 and v2 lie in distinct components of Si (M)and admit representatives whose intersection has i + 1components.

For all i , Si (M) is a flag complex.

The surface complex of M, denoted S(M), is defined to beSi0(M), where i0 is the smallest natural number such that Si0(M)is connected.

Jennifer Schultens Surfaces in Seifert fibered spaces

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Surface complexes of Seifert fibered spaces

Theorem

If M is a totally orientable Seifert fibered space with nonzero Eulernumber, then S(M) is isomorphic to the curve complex of Q.

Theorem

If M is a totally orientable Seifert fibered space with base orbifoldof genus 0 and Euler number 0, then S(M) contains a subcomplexisomorphic to the curve complex of the surface obtained from Q.Moreover, S(M) is contained in the cone on this subcomplex.

Corollary

(Special case) If M is a totally orientable Seifert fibered space withbase orbifold of genus 0, Euler number 0, and either 4 or 5exceptional fibers with identical invariants, then S(M) isisomorphic to the cone on the curve complex of Q.

Jennifer Schultens Surfaces in Seifert fibered spaces

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Surface complexes of Seifert fibered spaces

Theorem

If M is a totally orientable Seifert fibered space with Euler number0 and base orbifold of positive genus, then S(M) contains asubcomplex isomorphic to the curve complex of the surfaceobtained from Q. Moreover, Sd(M) is connected, for d the leastcommon multiple of α1, . . . , αk . In particular, S(M) = Sd(M).

Jennifer Schultens Surfaces in Seifert fibered spaces

Page 81: Surfaces in Seifert fibered spaces

Computations

π1(M) =< a1, b1, . . . , ag , bg , x1, . . . , xk , h |

h−bΠg1 [ai , bi ]Π

k1xi , [a1, h], [b1, h], . . . , [ag , h], [bg , h],

[x1, h], . . . , [xk , h], xα11 hβ1 , . . . , xαk

k hβk >

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Computations

H1(M) =< a1, b1, . . . , ag , bg , x1, . . . , xn, h | x1 + · · ·+ xn,

α1x1 + β1h, . . . , αnxn + βnh >

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Computations

Relations of the form αixi + βih yield relations between the xi s.E.g.:

α1 = 3, β1 = 2, α2 = 5, β2 = 3

9x1 + 6h = 10x2 + 6h

9(x1 − x2) = x2

So:

< x1, x2 | 9x1 = 10x2 >=< x1 − x2, x2 | 9(x1 − x2) = x2 >

=< x1 − x2 >= Z

Jennifer Schultens Surfaces in Seifert fibered spaces

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Computations

Substitution of this type are examples of standard proceduresinvolving Nielsen equivalence and the Euclidean algorithm. Nielsenequivalence oftenprovides a method for reducing the number ofgenerators.

This allows us to compute H1 explicitly:

H1(M) =< a1, b1, . . . , ag , bg , η >

Jennifer Schultens Surfaces in Seifert fibered spaces

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Gratitude

Thank you for listening and thanks to the organizers for invitingme to your conference!

Jennifer Schultens Surfaces in Seifert fibered spaces