Topological Symmetries of Molecules · Topological Symmetries of Molecules Erica Flapan December 13, 2013 ... Kuratowski cyclophane also has the underlying form of a molecular Mobius

Topological Symmetries of Molecules

Erica Flapan

December 13, 2013

Workshop: Topological Structures in Computational BiologyInstitute for Mathematics and its Applications

Erica Flapan Topological Symmetries of Molecules

Molecular symmetries

The symmetries of a molecule determine many important aspectsof its behavior.

For example, symmetry is useful for:

• Predicting reactions

• Crystallography

• Spectroscopy

• Quantum chemistry

• Analyzing the electron structure of a molecule

• Classifying molecules


Living organisms

Asymmetric molecules interact with one another like feet andshoes.

Asymmetric objects


Living organisms

Asymmetric molecules interact with one another like feet andshoes.

Asymmetric objects

Amino acids, sugars, and other molecules in living organisms areasymmetric.

DNA is different from

its mirror image

Hence we react differently to mirror forms of asymmetric molecules.


Pharmaceuticals

Some pharmaceuticals and their mirror images:

• Ibuprofen is an anti-inflamatory, but its mirror form is inert.

• Naproxen is an anti-inflamatory, but its mirror form is toxic.

• Darvon is a pain killer, but its mirror form is the coughsuppressant Novrad.


Pharmaceuticals

Some pharmaceuticals and their mirror images:

• Ibuprofen is an anti-inflamatory, but its mirror form is inert.

• Naproxen is an anti-inflamatory, but its mirror form is toxic.

• Darvon is a pain killer, but its mirror form is the coughsuppressant Novrad.

Drugs are synthesized in a 50:50 mix of mirror forms.

If a molecule has mirror image symmetry these are the same.

Otherwise, the two forms may need to be separated to avoiddangerous side effects.

Knowing whether a structure will have mirror symmetry is useful indrug design.


Mirror image symmetry

But what do we mean by mirror symmetry?.

Definition:

A molecule is said to be chemically chiral if it can not transformitself into its mirror image at room temperature. Otherwise, it issaid to be chemically achiral.

Note: This definition describes the behavior of a molecule not itstopology or geometry.


Mirror image symmetry

But what do we mean by mirror symmetry?.

Definition:

A molecule is said to be chemically chiral if it can not transformitself into its mirror image at room temperature. Otherwise, it issaid to be chemically achiral.

Note: This definition describes the behavior of a molecule not itstopology or geometry.

Definition:

A rigid object is said to be geometrically chiral if it cannot besuperimposed on its mirror image. Otherwise, it is said to begeometrically achiral.


Geometric vs.chemical achirality

If an object can be rigidly superimposed on its mirror image, thenit is chemically the same as it’s mirror image.

Geometrically

AchiralChemically

Achiral(the same as mirror

image as a rigid object)

(can transform itself

into its mirror image)


Geometric vs.chemical achirality

If an object can be rigidly superimposed on its mirror image, thenit is chemically the same as it’s mirror image.

Geometrically

AchiralChemically

Achiral(the same as mirror

image as a rigid object)

(can transform itself

into its mirror image)

Thus the set of geometrically achiral molecules is a subset of theset of chemically achiral molecules.

geometrically achiral

chemically achiral

? But is there a chemically achiral molecule

which is not geometrically achiral?

C

H

CH3

H

C

H3C

Cl Cl


Geometric vs.chemical chirality

This molecule (without the faces) was synthesized by Kurt Mislowin 1954 to show that geometric and chemical chirality are different.

NO2

O

NO2

O2N

OO2N

O

C C

O

left propeller right propeller

{ {Propellers are behind the screen. They turn simultaneously.



This molecule (without the faces) was synthesized by Kurt Mislowin 1954 to show that geometric and chemical chirality are different.

NO2

O

NO2

O2N

OO2N

O

C C

O

left propeller right propeller

{ {Propellers are behind the screen. They turn simultaneously.

Left propeller has her left hand forward, right propeller has herright hand forward. So, as rigid structures, a right propeller isdifferent from a left propeller.


Molecule is chemically achiral

NO

NO2

2

O2N

O2N

O

O

C

O

C

O

NO2

O

NO2

O2N

OO2N

O

C C

O

Original Mirror form



NO

NO2

2

O2N

O2N

O

O

C

O

C

O

NO2

O

NO2

O2N

OO2N

O

C C

O


Mirror form is the same as original, except vertical and horizontalhexagons have switched places.



NO

NO2

2

O2N

O2N

O

O

C

O

C

O

NO2

O

NO2

O2N

OO2N

O

C C

O



Proof of chemical achirality



NO

NO2

2

O2N

O2N

O

O

C

O

C

O

NO2

O

NO2

O2N

OO2N

O

C C

O




• Rotate original molecule by 90◦ about a horizontal axis to getmirror form with propellers horizontal.



NO

NO2

2

O2N

O2N

O

O

C

O

C

O

NO2

O

NO2

O2N

OO2N

O

C C

O




• Rotate original molecule by 90◦ about a horizontal axis to getmirror form with propellers horizontal.

• Rotate propellers back to vertical position to get mirror form.


Molecule is geometrically chiral

NO

NO2

2

O2N

O2N

O

O

C

O

C

O

NO2

O

NO2

O2N

OO2N

O

C C

O


Proof of geometric chirality

• Suppose molecule is rigid. Then propellers don’t rotate.

• In original form, left propeller is parallel to adjacent hexagon.

• In mirror form, left propeller is perpendicular to adjacenthexagon.

• A left propeller cannot change into a right propeller.

• As rigid objects, the original and mirror form are distinct.



This example shows that for non-rigid molecules, geometricchirality does not necessarily imply chemical chirality.

chemically achiral

NO2

O

NO2

O2N

OO2N

O

C C

O

geometrically

achiralC

H

CH3

H

C

H3C

Cl Cl


Molecular rigidity and non-rigidity

In fact, some molecules are rigid, some are flexible, and some havepieces that can rotate around certain bonds.

O

O

O OO

O

O

O

OO

O

O

OO

O

O

O

O

C

H

H

H

H

rigid flexible

Br

Cl

ClCl

H C3

rotating propeller

NO2

O

NO2

O2N

O

O2N

O

C C

O

two simultaneous propellersErica Flapan Topological Symmetries of Molecules

Topological Chirality

So no mathematical characterization of chemical chirality thatworks for all molecules is possible.

The definition of geometric chirality treats all molecules ascompletely rigid, which is not correct.

Now we will treat all molecules as completely flexible, which is alsonot correct.

The truth is somewhere in the middle.

Definition

A molecule is said to be topologically achiral if, assuming completeflexibility, it is isotopic to its mirror image. Otherwise it is said tobe topologically chiral.


Topological vs chemical chirality

If this molecule were flexible, we could grab the CO2H and push itto the left, while pulling the H to the right.

C

NH2

H

CH3

CO2H H

CH3

C

NH2

HO2C

mirrorinterchange

So it is topologically achiral. However, the molecule is rigid so it’schemically and geometrically chiral.

Topologically

achiralChemically

achiral

(the same as mirror image

as a flexible object)

(the same as mirror

image experimentally)


Topological chirality

Topologically chiralGeometrically chiral

(different from mirror

image as a rigid

object)




image as a flexible

object)

Chemically chiral

None of the reverse implications hold.


Topological chirality

Topologically chiralGeometrically chiral


image as a rigid

object)




image as a flexible

object)

Chemically chiral

None of the reverse implications hold.

If we heat a molecule which is geometrically chiral but nottopologically chiral, we can force it to change to its mirror form.

Even if we heat a topologically chiral molecule, it will not changeto its mirror form.

Thus knowing whether or not a molecule is topologically chiralhelps to predict its behavior.


The first example

In 1986, Jon Simon gave the first example of a topologically chiralmolecule by proving that a molecular Mobius ladder with threerungs is topologically chiral.

O

O

O OO

O

O

O

OO

O

O

OO

O

O

O

O

We sketch Simon’s proof.


Set-up

We represent the molecule as a colored graph M3, distinguishingsides from the rungs, because chemically they are different.

O

O

O OO

O

O

O

OO

O

O

OO

O

O

O

O

M3

The different rung colors help us keep track of each rung, but arenot meant to distinguish one from another.


Set-up

We represent the molecule as a colored graph M3, distinguishingsides from the rungs, because chemically they are different.

O

O

O OO

O

O

O

OO

O

O

OO

O

O

O

O

M3

The different rung colors help us keep track of each rung, but arenot meant to distinguish one from another.

Isotop sides of ladder to a planar circle A.

AErica Flapan Topological Symmetries of Molecules

2-fold branched covers

We obtain the 2-fold branched cover, by gluing two copies ofladder together along A.

A= p(fix(h))h fix(h)

p

S3M= S

3N=

quotient map

Formal definition

M, N = 3–manifolds, h : M → M orientation preservinghomeomorphism of order 2, and p : M → N quotient map inducedby h. If A = p(fix(h)) is a 1–manifold, then we say M is the 2–foldbranched cover of N branched over A.


Sketch of branched cover argument

Remove A to obtain a link L.

2-fold branched cover L

Use linking numbers to prove that the link L is topologically chiral.

If the Mobius ladder M3 were topologically achiral, then we couldlift the isotopy to get an isotopy taking L to its mirror image.

Thus the molecular Mobius ladder is topologically chiral,distinguishing the rungs from the sides.


Mobius strip

Maybe this is not surprising, since a Mobius strip is topologicallychiral.

mirror


Another Mobius ladder

Kuratowski cyclophane also has the underlying form of a molecularMobius ladder M3, though its molecular structure is quite different.

O

O O

O

O

O

O

O

O

O

O OO

O

O

O

OO

O

O

OO

O

O

O

O

Kuratowski cylcophane

Mobius ladderembedded

graph


An achiral Mobius ladder with three rungs

O

O O

O

O

O

O

O

O

O O

O

O

O

O

O

The black is in the plane of reflection, pink is in front, and blue isin back.

Thus Kuratowski cyclophane is geometrically achiral, though hasthe form of a Mobius ladder with three rungs.


A Mobius ladder with four rungs

We see as follows that this iron-sulfur cluster contains a Mobiusladder with four rungs.

[CH2]8

[CH2]8

[CH2]8

[CH2]8

N

N

N

N

R

S

Fe

R

S

Fe

S

S

SR

S

Fe

S

R

Fe

S

3

4

5

8

1

2

6

7

underlying structure


Mobius ladder with four rungs

The sides of the Mobius ladder are green and the rungs are pink.We ignore the black.

3

4

5

8

1

2

6

7

1

2

3

4

5

6

7

8


Structure is geometrically achiral

[CH2]8

[CH2]8

[CH2]8

NN

NN

R

SFe

R

S

Fe

S

S

S

R

S

Fe

SR

Fe

S

[CH2]8

[CH2]8

[CH2]8

[CH2]8

N

N

N

N

R

S

Fe

R

S

Fe

S

S

SR

S

Fe

S

R

Fe

S

mirror

rotate clockwise

[CH2]8

To get mirror image, rotate clockwise by 90◦.Erica Flapan Topological Symmetries of Molecules

Topological Chirality

Various methods have been developed to prove that graphsembedded in R3 are topologically chiral, whether or not they aremolecular graphs.

Theorem [Flapan]

If an abstract graph contains K5 or K3,3 and has no order 2automorphism, then any embedding of the graph in R3 istopologically chiral.

a b c

1 2 3K3,3

1

2 3

4

5

K5


Ferrocenophane

O

Feferrocenophane

fixed2

4

Any automorphism of a molecular graph must take atoms of agiven type to atoms of the same type.

Hence any automorphism of ferrocenophane fixes the oxygen, andhence fixes the adjacent vertex.

Since a automorphism cannot interchange vertices of differentvalence, it must also fix vertex 2 and vertex 4.


Ferrocenophane

O

Feferrocenophane

fixed

fixed

fixed

Now progressively we see that more and more vertices are fixed.


Ferrocenophane

O

Feferrocenophane

fixed

fixed

fixed

Now progressively we see that more and more vertices are fixed.

O

Feferrocenophane

fixed

fixed

fixed

fixedfixed

fixed

In fact, every vertex is fixed. So ferrocenophane has no non-trivialautomorphisms.


Ferrocenophane

To see ferrocenophane contains K5.

12

3

4

5

O

Fe

12

3

4

5K 5


Ferrocenophane

To see ferrocenophane contains K5.

12

3

4

5

O

Fe

12

3

4

5K 5

Thus, since it has no order 2 automorphism and contains K5,ferrocenophane is topologically chiral by the theorem.


Intrinsic chirality

Definition

A graph G is said to be intrinsically chiral, if every embedding of Gin space is topologically chiral.

That is, the chirality is intrinsic to the graph and does not dependon the particular embedding.


Intrinsic chirality

Definition

A graph G is said to be intrinsically chiral, if every embedding of Gin space is topologically chiral.

That is, the chirality is intrinsic to the graph and does not dependon the particular embedding.

Theorem [Flapan]

If an abstract graph contains K5 or K3,3 and has no order 2automorphism, then any embedding of the graph in R3 istopologically chiral.

Theorem Restatement

If an abstract graph contains K5 or K3,3 and has no order 2automorphism, then the graph is intrinsically chiral.


Intrinsically chirality

Theorem Restatement

If an abstract graph contains K5 or K3,3 and has no order 2automorphism, then the graph is intrinsically chiral.

In chemical terms, if a molecule is intrinsically chiral then it and allof its stereoisomers are topologically chiral.

O

Feferrocenophane

Thus ferrocenophane is intrinsically chiral.


Topological chirality does not imply intrinsically chirality

These are different embeddings of the same molecular graph.

O

OO O O

O

O

OO

O O OO

O

N N N N

N N N N

OO

OO

O

O

O

O

O

OO O

O

O

N

N

N

N

N

N

N

N


Topological chirality does not imply intrinsically chirality

These are different embeddings of the same molecular graph.

O

OO O O

O

O

OO

O O OO

O

N N N N

N N N N

OO

OO

O

O

O

O

O

OO O

O

O

N

N

N

N

N

N

N

N

The knotted molecule is topologically chiral (because a trefoil knotis topologically chiral), and the unknotted molecule is topologicallyachiral (because it’s planar).

Thus the knotted molecule is topologically chiral but notintrinsically chiral.


Hierarchy of chirality

Topologically chiral

Geometrically chiral


image as a rigid

object)




image as a flexible

object)Chemically chiral

Intrinsically chiral

(all embeddings different

from mirror image as a

flexible object)

O

OO

O OO

O

O OO O O

O

ON N

H

CH3

C

NH2

HO2C

NO2

O

NO2

O2N

OO2N

O

C C

O

ON N

N NN N

Fe

None of the reverse implications hold.Erica Flapan Topological Symmetries of Molecules

Other types of symmetries

Mirror image symmetry is not the only type of molecular symmetrythat is chemically significant.

Chemists define

The point group of a molecular graph as its group of rotations,reflections, and reflections composed with rotations.

It’s called the point group because it fixes a point of R3.

Chemists use the point group to classify molecules.


Non-rigid molecules

Like geometry chirality, the point group treats all molecules as ifthey are rigid. But as we saw, not all molecules are rigid.

The top of this molecule spins like a propeller.

Br

Cl

Cl

Cl

H C3


Non-rigid molecules

Like geometry chirality, the point group treats all molecules as ifthey are rigid. But as we saw, not all molecules are rigid.

The top of this molecule spins like a propeller.

Br

Cl

Cl

Cl

H C3

Planar reflection pointwise fixing the three hexagons is its onlyrigid symmetry. So point group is Z2.

We would like a symmetry group that includes the reflection aswell as an order 3 rotation of the propeller.


Molecular symmetry group

Definition

Let Γ be a molecular graph, and let Aut(Γ) denote the group ofautomorphisms of Γ taking atoms of a given type to atoms of thesame type. The molecular symmetry group of Γ is the subgroup ofAut(Γ) induced by chemically possible motions taking Γ to itself orits reflection.



Definition

Let Γ be a molecular graph, and let Aut(Γ) denote the group ofautomorphisms of Γ taking atoms of a given type to atoms of thesame type. The molecular symmetry group of Γ is the subgroup ofAut(Γ) induced by chemically possible motions taking Γ to itself orits reflection.

Cl Br

12

3

The molecular symmetry group is 〈(12), (123)〉.



Cl Br

1

2

3

circle

Molecular symmetry group induces an isomorphic action on thecircle at the top.

1

2

3

D = <(123),(23)>=

dihedral group with 6 elements3

Molecular symmetry group = D3 = Aut(Γ)


Geometry vs topology

Definition

The topological symmetry group TSG(Γ) of a molecular graph Γ,is the subgroup of Aut(Γ) induced by homeomorphisms of R3

taking atoms of a given type to atoms of the same type.

Analogous to what we saw with achirality

• The point group treats all molecules as completely rigid.

• The topological symmetry group treats all molecules ascompletely flexible.

• The truth is somewhere in the middle.


Topological symmetry groups

Cl Br

1

2

3

circle

For this molecule, TSG(Γ) = molecular symmetry group = D3.

Point group 6=molecular symmetry group.


Topological symmetry groups

Cl Br

1

2

3

circle

For this molecule, TSG(Γ) = molecular symmetry group = D3.

Point group 6=molecular symmetry group.

Another example:

O

O

O OO

O

O

O

OO

O

O

OO

O

O

O

O


Molecular Mobius Ladder

We represent molecule as a colored graph where automorphismsmust preserve colors.

1

2

3

4

5

6O

O

O OO

O

O

O

OO

O

O

OO

O

O

O

O

(23)(56)(14) is the only automorphism induced by a rigid motion.So point group = Z2.


Molecular Mobius Ladder

We represent molecule as a colored graph where automorphismsmust preserve colors.

1

2

3

4

5

6O

O

O OO

O

O

O

OO

O

O

OO

O

O

O

O

(23)(56)(14) is the only automorphism induced by a rigid motion.So point group = Z2.

(123456) is induced by rotating the molecule by 120◦ whileslithering the half-twist back to its original position.

TSG(Γ) = Molecular symmetry group = 〈(23)(56)(14), (123456)〉

So point group 6= Molecular symmetry group.


Different types of symmetry groups

⊆⊆ ⊆Point

Group

Molecular

Symmetry

Group

Topological

Symmetry

Group

Automorphism

Group

automorphisms

of the abstract

graph

automorphisms

induced by

molecular

motions

automorphisms

induced by

rotations

and

reflections

automorphisms

induced by

homeomorphisms

of space

Note

The point group is normally defined in terms of rotations andreflections of R3 rather than in terms of automorphisms. We writeit this way to compare it to the other types of symmetry groups.


Arbitrary Graphs embedded in S3

While motivated by molecular symmetries, TSG(Γ) can be definedfor any graph Γ embedded in R3.

Embedded graphs are a natural extension of knot theory, since wecan put vertices on a knot to make it into an embedded graph.

Symmetries are nicer in S3 = R3 ∪ {∞} than in R3.

1 2

The ends of this knot are attached in S3 and (12) is induced by arotation-reflection. If the ends are attached in R3, the symmetry isnot so nice.

1 2


Topological Symmetry Groups

Definition

The topological symmetry group of a graph Γ embedded in S3,TSG(Γ), is the subgroup of Aut(Γ) induced by homeomorphismsof (S3, Γ).


Topological Symmetry Groups

Definition

The topological symmetry group of a graph Γ embedded in S3,TSG(Γ), is the subgroup of Aut(Γ) induced by homeomorphismsof (S3, Γ).

Frucht proved that every finite group is isomorphic to Aut(Γ) forsome graph Γ.

Is every finite group isomorphic to TSG(Γ) for some graph Γembedded in S3?

Before answering, we illustrate some groups that are topologicalsymmetry groups.


What groups can be TSG(Γ)?

Γ

chiral

Wheels can rotate but can’t be interchanged.

TSG(Γ) = Z2 × Z3 × Z4


Any finite abelian group

We can have any number of wheels with any number of spokes.

If two wheels have the same number of spokes, we can add distinct(chiral) knots so the wheels can’t be interchanged.


Any finite abelian group

We can have any number of wheels with any number of spokes.

If two wheels have the same number of spokes, we can add distinct(chiral) knots so the wheels can’t be interchanged.

Γ

TSG(Γ) = Z2 × Z2 × Z4

In this way, any finite abelian group can be TSG(Γ).


Symmetric groups

v

w

1 2 n

non-invertible

chiral

w

v

w

vrotated flipped

over

A non-invertible knot is one that is not isotopic to itself with itsorientation reversed.


Symmetric groups

v

w

1 2 n

non-invertible

chiral

w

v

w

vrotated flipped

over

A non-invertible knot is one that is not isotopic to itself with itsorientation reversed.

Non-invertible & chiral knots ⇒ no homeomorphism induces (vw).

Any transposition (ij) is induced by twisting strands.

Thus TSG(Γ) = Sn, the group of permutations of n points.


What about alternating groups?

v

w

1 2 n

non-invertible

chiral

Can we get TSG(Γ) = An by adding different knots?

Can we get TSG(Γ) = An for another embedded graph Γ?


What about alternating groups?

v

w

1 2 n

non-invertible

chiral

Can we get TSG(Γ) = An by adding different knots?

Can we get TSG(Γ) = An for another embedded graph Γ?

Not unless n ≤ 5.

Theorem [Flapan, Naimi, Pommersheim, Tamvakis]

TSG(Γ) can be An for some graph Γ embedded in S3 iff n ≤ 5.


TSG+(Γ)

Definition

TSG+(Γ) is the subgroup of TSG(Γ) induced by orientationpreserving homeomorphisms of S3.

TSG +(Γ)= Ζ2× Ζ3

× Ζ4TSG +

(Γ)= Ζ2× Ζ3

× Ζ4

TSG (Γ)= Ζ2× Ζ3

× Ζ4( ) Ζ2

TSG (Γ)= Ζ2× Ζ3

× Ζ4


Finite order homeomorphisms

TSG+(Γ) = either TSG(Γ) or a normal subgroup of index 2.

So studying TSG+(Γ) is almost as good as studying TSG(Γ), butit’s simpler.


Finite order homeomorphisms

TSG+(Γ) = either TSG(Γ) or a normal subgroup of index 2.

So studying TSG+(Γ) is almost as good as studying TSG(Γ), butit’s simpler.

A function f has finite order if for some n > 0, f n is the identity.

All automorphisms in TSG+(Γ) have finite order, but are theyinduced by finite order homeomorphisms of S3?

Consider what happens to the red arc when we spin a wheel.

12

3

1

2

3

1

23

12

3


Homeomorphisms of (S3, Γ) may not have finite order.

The wheel returns to its original position, but the red arc does not.

12

3

1

2

3

1

23

12

3

Spinning a wheel has finite order on Γ, but not on S3.

Г

TSG+(Γ) is not induced by a finite group of homeomorphisms of(S3, Γ).

But, this is a special (bad) type of graph.


3-connected graphs

Definition

An abstract graph γ is 3-connected if at least 3 vertices togetherwith their edges must be removed in order to disconnect γ orreduce it to a single vertex.


3-connected graphs

Definition

An abstract graph γ is 3-connected if at least 3 vertices togetherwith their edges must be removed in order to disconnect γ orreduce it to a single vertex.

v

w

1 2 n

removered verticesto disconnectgraphs

Neither of these graphs is 3-connected.


A 3-connected graph

1

3

4

5 62

3-connected

(56)(23) is induced by turning the graph over.

(153426) is induced by slithering the graph along itself whileinterchanging the pink and blue knots.


A 3-connected graph

1

3

4

5 62

3-connected

(56)(23) is induced by turning the graph over.

(153426) is induced by slithering the graph along itself whileinterchanging the pink and blue knots.

(153426) is not induced by a finite order homeomorphism of S3

because there is no order 6 homeomorphism of S3 taking the figureeight knot to itself, and no knot can be the fixed point set of afinite order homeomorphism.

TSG+(Γ) = 〈(56)(23), (153426)〉 = D6. But TSG+(Γ) is notinduced by a finite group of homeomorphisms.


A nicer embedding of Γ

Here is another embedding of Γ, where the same automorphismsare induced by finite order homeomorphisms.

Γ

1

4

52

63

Γ

1

4

3 65 2re-embed

(56)(23) is induced by turning Γ′ over left to right.

(153426) is induced by a glide rotation of Γ′ that interchanges thetwo circles while rotating counterclockwise. This glide rotation hasfinite order.

TSG+(Γ′) = D6 is induced by an isomorphic finite group of

homeomorphisms of S3.Erica Flapan Topological Symmetries of Molecules

Isometries

Finiteness Theorem [Flapan, Naimi, Pommersheim,Tamvakis]

Any 3-connected graph Γ embedded in S3 can be re-embedded asΓ′ so that TSG+(Γ) is a subgroup of TSG+(Γ

′) and is induced bya finite group of isometries of S3.


Isometries




1

23

4

56

1

23

4

56

<(123)(456), (23)(56)> <(123)(456), (23)(56), (14)(25)(36)>

Γ Γ

TSG ( )=Γ +TSG ( )=Γ +

(23)(56) is induced by a finite order homeomorphism of (S3, Γ′)but not of (S3, Γ). But TSG+(Γ

′) TSG+(Γ).


Embeddings of graphs in other manifolds

Definition

Let Γ be a graph embedded in a 3-dimensional manifold M, thenTSG(Γ,M) is the subgroup of Aut(Γ) induced byhomeomorphisms of M.


Embeddings of graphs in other manifolds

Definition

Let Γ be a graph embedded in a 3-dimensional manifold M, thenTSG(Γ,M) is the subgroup of Aut(Γ) induced byhomeomorphisms of M.

Theorem [Flapan,Tamvakis]

Let M be a closed (i.e., compact and without boundary),connected, orientable, irreducible (i.e., can’t be split along spheresinto simpler manifolds) 3-manifold. Then there exists a finitesimple group which is not isomorphic to TSG(Γ,M) for any graphΓ embedded in M.

By our earlier result if M = S3, then G = An where n > 5.


All finite groups are possible if M varies


Let M be a closed, connected, orientable, irreducible 3-manifold.Then there exists a finite simple group which is not isomorphic toTSG(Γ,M) for any graph Γ embedded in M.

The above theorem is for a fixed 3-manifold M. If we allow M tovary, then every finite group can occur.


For every finite group G , there is a 3-connected graph Γ embeddedin some 3-manifold M such that TSG(Γ,M) ∼= G . In fact, M canbe chosen to be a hyperbolic rational homology sphere.


Finiteness Theorem

Recall that for S3 we had:




Note that S3 is Seifert fibered.


Finiteness Theorem

Recall that for S3 we had:




Note that S3 is Seifert fibered.

Non-Finiteness Theorem [Flapan, Tamvakis]

For every closed, orientable, irreducible, 3-manifold M which is notSeifert fibered, there is a 3-connected graph Γ embedded in M

such that TSG+(Γ,M) is not isomorphic to any finite group ofhomeomorphisms of M.


Thanks

H

TA

SN

K


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