Topological Symmetries of Molecules

Home , Chirality (mathematics), Symmetry group

Erica Flapan

December 13, 2013

Workshop: Topological Structures in Computational Biology Institute for Mathematics and its Applications

Erica Flapan Topological Symmetries of Molecules Molecular symmetries

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

For example, symmetry is useful for:

• Predicting reactions • Crystallography • Spectroscopy • Quantum chemistry • Analyzing the electron structure of a molecule • Classifying molecules

Erica Flapan Topological Symmetries of Molecules Living organisms

Asymmetric molecules interact with one another like feet and shoes.

Asymmetric objects

Erica Flapan Topological Symmetries of Molecules Living organisms

Asymmetric molecules interact with one another like feet and shoes.

Asymmetric objects

Amino acids, sugars, and other molecules in living organisms are asymmetric.

DNA is different from its mirror image

Hence we react diﬀerently to mirror forms of asymmetric molecules.

Erica Flapan Topological Symmetries of 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 cough suppressant Novrad.

Erica Flapan Topological Symmetries of Molecules Pharmaceuticals

Some pharmaceuticals and their mirror images:

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 avoid dangerous side eﬀects. Knowing whether a structure will have mirror symmetry is useful in drug design.

Erica Flapan Topological Symmetries of Molecules Mirror image symmetry

But what do we mean by mirror symmetry?.

Deﬁnition: A molecule is said to be chemically chiral if it can not transform itself into its mirror image at room temperature. Otherwise, it is said to be chemically achiral.

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

Erica Flapan Topological Symmetries of Molecules Mirror image symmetry

But what do we mean by mirror symmetry?.

Deﬁnition: A molecule is said to be chemically chiral if it can not transform itself into its mirror image at room temperature. Otherwise, it is said to be chemically achiral.

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

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

Erica Flapan Topological Symmetries of Molecules Geometric vs.chemical achirality

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

Geometrically Chemically Achiral Achiral (the same as mirror (can transform itself image as a rigid object) into its mirror image)

Erica Flapan Topological Symmetries of Molecules Geometric vs.chemical achirality

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

Geometrically Chemically Achiral Achiral (the same as mirror (can transform itself image as a rigid object) into its mirror image)

Thus the set of geometrically achiral molecules is a subset of the set of chemically achiral molecules. chemically achiral ? But is there a chemically achiral molecule which is not geometrically achiral ? geometrically achiral Cl Cl

C C H 3C CH 3 H H

Erica Flapan Topological Symmetries of Molecules Geometric vs.chemical chirality

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

NO 2

O2N O O C C O O O2N { {

left propellerNO 2 right propeller

Propellers are behind the screen. They turn simultaneously.

Erica Flapan Topological Symmetries of Molecules Geometric vs.chemical chirality

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

NO 2

O2N O O C C O O O2N { {

left propellerNO 2 right propeller

Propellers are behind the screen. They turn simultaneously.

Left propeller has her left hand forward, right propeller has her right hand forward. So, as rigid structures, a right propeller is diﬀerent from a left propeller. Erica Flapan Topological Symmetries of Molecules Molecule is chemically achiral

Original Mirror form NO 2 O2N

O2N O O NO 2 O O C C C C O O O O O2N NO 2

NO2 O2N

Erica Flapan Topological Symmetries of Molecules Molecule is chemically achiral

Original Mirror form NO 2 O2N

O2N O O NO 2 O O C C C C O O O O O2N NO 2

NO2 O2N

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

Erica Flapan Topological Symmetries of Molecules Molecule is chemically achiral

Original Mirror form NO 2 O2N

O2N O O NO 2 O O C C C C O O O O O2N NO 2

NO2 O2N

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

Proof of chemical achirality

Erica Flapan Topological Symmetries of Molecules Molecule is chemically achiral

Original Mirror form NO 2 O2N

O2N O O NO 2 O O C C C C O O O O O2N NO 2

NO2 O2N

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

Proof of chemical achirality

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

Erica Flapan Topological Symmetries of Molecules Molecule is chemically achiral

Original Mirror form NO 2 O2N

O2N O O NO 2 O O C C C C O O O O O2N NO 2

NO2 O2N

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

Proof of chemical achirality

• Rotate original molecule by 90◦ about a horizontal axis to get mirror form with propellers horizontal. • Rotate propellers back to vertical position to get mirror form.

Erica Flapan Topological Symmetries of Molecules Molecule is geometrically chiral

Original Mirror form NO 2 O2N

O2N O O NO 2 O O C C C C O O O O O2N NO 2

NO2 O2N

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 adjacent hexagon. • A left propeller cannot change into a right propeller. • As rigid objects, the original and mirror form are distinct.

Erica Flapan Topological Symmetries of Molecules Geometric vs.chemical chirality

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

chemically achiral

NO 2

O2N O O C C O O O2N

NO 2 Cl Cl geometrically C C achiral H 3C CH 3 H H

Erica Flapan Topological Symmetries of Molecules Molecular rigidity and non-rigidity

In fact, some molecules are rigid, some are ﬂexible, and some have pieces that can rotate around certain bonds. Cl Cl Cl H O O O O O O Br C O O H 3 C O O H H O O O O O O H O O rigid flexible rotating propeller

NO 2

O2N O O C C O O O2N

NO 2 two simultaneous propellers

Erica Flapan Topological Symmetries of Molecules Topological Chirality

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

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

Now we will treat all molecules as completely ﬂexible, which is also not correct.

The truth is somewhere in the middle.

Deﬁnition A molecule is said to be topologically achiral if, assuming complete ﬂexibility, it is isotopic to its mirror image. Otherwise it is said to be topologically chiral.

Erica Flapan Topological Symmetries of Molecules Topological vs chemical chirality

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

CH 3 CH 3

C C

H H CO 2 H HO 2C

NH 2 NH 2

interchange mirror So it is topologically achiral. However, the molecule is rigid so it’s chemically and geometrically chiral.

Topologically Chemically achiral achiral (the same as mirror image (the same as mirror as a flexible object) image experimentally)

Erica Flapan Topological Symmetries of Molecules Topological chirality

Topologically chiral Chemically chiral Geometrically chiral (different from mirror (different from mirror image as a flexible (different from mirror image experimentally) image as a rigid object) object)

None of the reverse implications hold.

Erica Flapan Topological Symmetries of Molecules Topological chirality

None of the reverse implications hold.

If we heat a molecule which is geometrically chiral but not topologically chiral, we can force it to change to its mirror form.

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

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

Erica Flapan Topological Symmetries of Molecules The ﬁrst example

In 1986, Jon Simon gave the ﬁrst example of a topologically chiral molecule by proving that a molecular M¨obius ladder with three rungs is topologically chiral.

O O O O O O O O O O O O O O O O O O

We sketch Simon’s proof.

Erica Flapan Topological Symmetries of Molecules Set-up

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

O O O O O O O O O O O O M 3 O O O O O O

The diﬀerent rung colors help us keep track of each rung, but are not meant to distinguish one from another.

Erica Flapan Topological Symmetries of Molecules Set-up

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

O O O O O O O O O O O O M 3 O O O O O O

The diﬀerent rung colors help us keep track of each rung, but are not meant to distinguish one from another.

Isotop sides of ladder to a planar circle A.

Erica Flapan Topological Symmetries of Molecules 2-fold branched covers

We obtain the 2-fold branched cover, by gluing two copies of ladder together along A. 3 M= S 3 N= S

p quotient map h fix(h) A= p(fix(h))

Formal deﬁnition M, N = 3–manifolds, h : M → M orientation preserving homeomorphism of order 2, and p : M → N quotient map induced by h. If A = p(ﬁx(h)) is a 1–manifold, then we say M is the 2–fold branched cover of N branched over A.

Erica Flapan Topological Symmetries of Molecules 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 M¨obius ladder M3 were topologically achiral, then we could lift the isotopy to get an isotopy taking L to its mirror image.

Thus the molecular M¨obius ladder is topologically chiral, distinguishing the rungs from the sides.

Erica Flapan Topological Symmetries of Molecules M¨obius strip

Maybe this is not surprising, since a M¨obius strip is topologically chiral.

mirror

Erica Flapan Topological Symmetries of Molecules Another M¨obius ladder

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

O O O O O O O O O O O O O O O O O O O O O O O O O Mobius ladder O embedded graph Kuratowski cylcophane

Erica Flapan Topological Symmetries of Molecules An achiral M¨obius ladder with three rungs

O O O O O O O O

The black is in the plane of reflection, pink is in front, and blue is in back. Thus Kuratowski cyclophane is geometrically achiral, though has the form of a Möbius ladder with three rungs. Erica Flapan Topological Symmetries of Molecules A Möbius ladder with four rungs

We see as follows that this iron-sulfur cluster contains a M¨obius ladder with four rungs.

[CH 2]8 N 3 7 N S R R S 4 Fe S S Fe 8 [CH 2]8 [CH ] Fe 2 8 S S 5 R Fe S R 1 6 S N N

[CH 2]8 2 underlying structure

Erica Flapan Topological Symmetries of Molecules M¨obius ladder with four rungs

The sides of the M¨obius ladder are green and the rungs are pink. We ignore the black.

3 7

3 7 4 4 8

8 5 5 1 1 6

2 6 2

Erica Flapan Topological Symmetries of Molecules Structure is geometrically achiral

[CH ] ] 2 8 [CH 2 8 N N N R N S R R R S S Fe S Fe S S S Fe [CH ] [CH ] 2 8 ] Fe S 2 8 [CH ] Fe [CH 2 8 2 8 S S Fe S R S R S Fe Fe R S N N R S S N N ] m [CH 2 8 irror [CH 2]8

rotate clockwise

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

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

Theorem [Flapan]

If an abstract graph contains K5 or K3,3 and has no order 2 automorphism, then any embedding of the graph in R3 is topologically chiral. 5 a b c

1 4

2 3 1 2 3 K K3,3 5 Erica Flapan Topological Symmetries of Molecules Ferrocenophane

ferrocenophane Fe

2 fixed 4 O Any automorphism of a molecular graph must take atoms of a given type to atoms of the same type.

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

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

Erica Flapan Topological Symmetries of Molecules Ferrocenophane

ferrocenophane Fe

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

Erica Flapan Topological Symmetries of Molecules Ferrocenophane

ferrocenophane Fe

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

ferrocenophane Fe fixed fixed fixed fixed fixed fixed O In fact, every vertex is ﬁxed. So ferrocenophane has no non-trivial automorphisms.

Erica Flapan Topological Symmetries of Molecules Ferrocenophane

To see ferrocenophane contains K5.

1 2 1 2

3 3 K 5 Fe 5 5

4 4

Erica Flapan Topological Symmetries of Molecules Ferrocenophane

To see ferrocenophane contains K5.

1 2 1 2

3 3 K 5 Fe 5 5

4 4

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

Erica Flapan Topological Symmetries of Molecules Intrinsic chirality

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

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

Erica Flapan Topological Symmetries of Molecules Intrinsic chirality

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

That is, the chirality is intrinsic to the graph and does not depend on the particular embedding. Theorem [Flapan]

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

Theorem Restatement

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

Erica Flapan Topological Symmetries of Molecules Intrinsically chirality

Theorem Restatement

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

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

ferrocenophane Fe

Thus ferrocenophane is intrinsically chiral.

Erica Flapan Topological Symmetries of Molecules Topological chirality does not imply intrinsically chirality

These are diﬀerent embeddings of the same molecular graph.

O O O O O O O O O O O O O N O N N N N N N N N N N N N O N O O O N O O O N O O O O O O O

Erica Flapan Topological Symmetries of Molecules Topological chirality does not imply intrinsically chirality

These are diﬀerent embeddings of the same molecular graph.

O O O O O O O O O O O O O N O N N N N N N N N N N N N O N O O O N O O O N O O O O O O O

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

Thus the knotted molecule is topologically chiral but not intrinsically chiral. Erica Flapan Topological Symmetries of Molecules Hierarchy of chirality

Intrinsically chiral (all embeddings different from mirror image as a flexible object) Topologically chiral (different from mirror image as a flexible object) Fe Chemically chiral O O O O O (different from mirror O O image experimentally) N N N N N N N N CH 3 Geometrically chiral O O (different from mirror O O C O O O O image as a rigid H HO 2C object) NH 2

NO 2 O N O 2 O C C O O O2N NO 2 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 symmetry that 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.

Erica Flapan Topological Symmetries of Molecules Non-rigid molecules

Like geometry chirality, the point group treats all molecules as if they are rigid. But as we saw, not all molecules are rigid. The top of this molecule spins like a propeller. Cl Cl Cl

Br H 3 C

Erica Flapan Topological Symmetries of Molecules Non-rigid molecules

Like geometry chirality, the point group treats all molecules as if they are rigid. But as we saw, not all molecules are rigid. The top of this molecule spins like a propeller. Cl Cl Cl

Br H 3 C

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

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

Erica Flapan Topological Symmetries of Molecules Molecular symmetry group

Deﬁnition Let Γ be a molecular graph, and let Aut(Γ) denote the group of automorphisms of Γ taking atoms of a given type to atoms of the same type. The molecular symmetry group of Γ is the subgroup of Aut(Γ) induced by chemically possible motions taking Γ to itself or its reﬂection.

Erica Flapan Topological Symmetries of Molecules Molecular symmetry group

2 3 1

Cl Br

The molecular symmetry group is h(12), (123)i.

Erica Flapan Topological Symmetries of Molecules Molecular symmetry group

circle 2 3 1

Cl Br

Molecular symmetry group induces an isomorphic action on the circle at the top. 2 D = <(123),(23)>= 1 3 dihedral group with 6 elements 3

Molecular symmetry group = D3 = Aut(Γ)

Erica Flapan Topological Symmetries of Molecules 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 as completely ﬂexible. • The truth is somewhere in the middle.

Erica Flapan Topological Symmetries of Molecules Topological symmetry groups

circle 2 3 1

Cl Br

For this molecule, TSG(Γ) = molecular symmetry group = D3. Point group 6=molecular symmetry group.

Erica Flapan Topological Symmetries of Molecules Topological symmetry groups

circle 2 3 1

Cl Br

For this molecule, TSG(Γ) = molecular symmetry group = D3. Point group 6=molecular symmetry group.

Another example: O O O O O O O O O O O O O O O O O O

Erica Flapan Topological Symmetries of Molecules Molecular M¨obius Ladder

We represent molecule as a colored graph where automorphisms must preserve colors.

O O O O 1 O O O O 2 O O 4 6 O O O 3 O O O 5 O O

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

Erica Flapan Topological Symmetries of Molecules Molecular M¨obius Ladder

We represent molecule as a colored graph where automorphisms must preserve colors.

O O O O 1 O O O O 2 O O 4 6 O O O 3 O O O 5 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◦ while slithering the half-twist back to its original position. TSG(Γ) = Molecular symmetry group = h(23)(56)(14), (123456)i So point group 6= Molecular symmetry group.

Erica Flapan Topological Symmetries of Molecules Diﬀerent types of symmetry groups

Molecular Topological Point ⊆ ⊆ ⊆ Automorphism Symmetry Symmetry Group Group Group Group automorphisms automorphisms automorphisms automorphisms induced by induced by induced by of the abstract rotations molecular homeomorphisms graph and motions of space reflections

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

Erica Flapan Topological Symmetries of Molecules Arbitrary Graphs embedded in S 3

While motivated by molecular symmetries, TSG(Γ) can be deﬁned for any graph Γ embedded in R3. Embedded graphs are a natural extension of knot theory, since we can 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 a rotation-reﬂection. If the ends are attached in R3, the symmetry is not so nice.

1 2

Erica Flapan Topological Symmetries of Molecules 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 homeomorphisms of (S3, Γ).

Erica Flapan Topological Symmetries of Molecules 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 homeomorphisms of (S3, Γ).

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

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

Before answering, we illustrate some groups that are topological symmetry groups.

Erica Flapan Topological Symmetries of Molecules What groups can be TSG(Γ)?

chiral

Wheels can rotate but can’t be interchanged.

TSG(Γ) = Z2 × Z3 × Z4

Erica Flapan Topological Symmetries of Molecules 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.

Erica Flapan Topological Symmetries of Molecules 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.

TSG(Γ) = Z2 × Z2 × Z4

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

Erica Flapan Topological Symmetries of Molecules Symmetric groups

w w non-invertible chiral

1 2 n rotated v v flipped over w

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

Erica Flapan Topological Symmetries of Molecules Symmetric groups

w w non-invertible chiral

1 2 n rotated v v flipped over w

A non-invertible knot is one that is not isotopic to itself with its orientation 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.

Erica Flapan Topological Symmetries of Molecules What about alternating groups?

non-invertible chiral

1 2 n

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

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

Erica Flapan Topological Symmetries of Molecules What about alternating groups?

non-invertible chiral

1 2 n

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

Erica Flapan Topological Symmetries of Molecules TSG+(Γ)

Deﬁnition

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

TSG (Γ)= × × TSG (Γ)= × × + Ζ 2 Ζ 3 Ζ 4 + Ζ 2 Ζ 3 Ζ 4

TSG (Γ)= × × TSG (Γ)= ( × × ) Ζ 2 Ζ 3 Ζ 4 Ζ 2 Ζ 3 Ζ 4 Ζ 2

Erica Flapan Topological Symmetries of Molecules 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), but it’s simpler.

Erica Flapan Topological Symmetries of Molecules 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), but it’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 they induced by ﬁnite order homeomorphisms of S3? Consider what happens to the red arc when we spin a wheel.

1 3 2 2 1 1 3 2 3 2 1 3

Erica Flapan Topological Symmetries of Molecules Homeomorphisms of (S 3, Γ) may not have ﬁnite order.

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

1 3 2 2 1 1 3 2 3 2 1 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.

Erica Flapan Topological Symmetries of Molecules 3-connected graphs

Deﬁnition An abstract graph γ is 3-connected if at least 3 vertices together with their edges must be removed in order to disconnect γ or reduce it to a single vertex.

Erica Flapan Topological Symmetries of Molecules 3-connected graphs

Deﬁnition An abstract graph γ is 3-connected if at least 3 vertices together with their edges must be removed in order to disconnect γ or reduce it to a single vertex.

v remove red vertices to disconnect graphs

1 2 n

w Neither of these graphs is 3-connected.

Erica Flapan Topological Symmetries of Molecules A 3-connected graph 1 4 3-connected 5 2 3 6

(56)(23) is induced by turning the graph over. (153426) is induced by slithering the graph along itself while interchanging the pink and blue knots.

Erica Flapan Topological Symmetries of Molecules A 3-connected graph 1 4 3-connected 5 2 3 6

(56)(23) is induced by turning the graph over. (153426) is induced by slithering the graph along itself while interchanging 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 figure eight knot to itself, and no knot can be the fixed point set of a finite order homeomorphism.

TSG+(Γ) = h(56)(23), (153426)i = D6. But TSG+(Γ) is not induced by a ﬁnite group of homeomorphisms.

Erica Flapan Topological Symmetries of Molecules A nicer embedding of Γ

Here is another embedding of Γ, where the same automorphisms are induced by ﬁnite order homeomorphisms. 1 1 4 Γ Γ 4 re-embed 5 2 3 6 5 6 2 3

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

(153426) is induced by a glide rotation of Γ′ that interchanges the two circles while rotating counterclockwise. This glide rotation has ﬁnite order. ′ TSG+(Γ )= D6 is induced by an isomorphic ﬁnite group of homeomorphisms of S3. Erica Flapan Topological Symmetries of Molecules Isometries

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 as ′ ′ Γ so that TSG+(Γ) is a subgroup of TSG+(Γ ) and is induced by a ﬁnite group of isometries of S3. 1 1 Γ Γ 4 4 6 6 5 5 2 2 3 3 TSG ( )= TSG + ( Γ )= + Γ <(123)(456), (23)(56)> <(123)(456), (23)(56), (14)(25)(36)> (23)(56) is induced by a ﬁnite order homeomorphism of (S3, Γ′) 3 ′ but not of (S , Γ). But TSG+(Γ ) TSG+(Γ).

Erica Flapan Topological Symmetries of Molecules Embeddings of graphs in other manifolds

Deﬁnition Let Γ be a graph embedded in a 3-dimensional manifold M, then TSG(Γ, M) is the subgroup of Aut(Γ) induced by homeomorphisms of M.

Erica Flapan Topological Symmetries of Molecules Embeddings of graphs in other manifolds

Deﬁnition Let Γ be a graph embedded in a 3-dimensional manifold M, then TSG(Γ, M) is the subgroup of Aut(Γ) induced by homeomorphisms 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 spheres into simpler manifolds) 3-manifold. Then there exists a ﬁnite simple group which is not isomorphic to TSG(Γ, M) for any graph Γ embedded in M.

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

Erica Flapan Topological Symmetries of Molecules All ﬁnite groups are possible if M varies

Theorem [Flapan,Tamvakis] Let M be a closed, connected, orientable, irreducible 3-manifold. Then there exists a ﬁnite simple group which is not isomorphic to TSG(Γ, M) for any graph Γ embedded in M.

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

Theorem [Flapan,Tamvakis] For every ﬁnite group G, there is a 3-connected graph Γ embedded in some 3-manifold M such that TSG(Γ, M) =∼ G. In fact, M can be chosen to be a hyperbolic rational homology sphere.

Erica Flapan Topological Symmetries of Molecules Finiteness Theorem

Recall that for S3 we had: Finiteness Theorem [Flapan, Naimi, Pommersheim,Tamvakis] Any 3-connected graph Γ embedded in S3 can be re-embedded as ′ ′ Γ so that TSG+(Γ) is a subgroup of TSG+(Γ ) and is induced by a ﬁnite group of isometries of S3.

Note that S3 is Seifert ﬁbered.

Erica Flapan Topological Symmetries of Molecules Finiteness Theorem

Note that S3 is Seifert ﬁbered.

Non-Finiteness Theorem [Flapan, Tamvakis] For every closed, orientable, irreducible, 3-manifold M which is not Seifert ﬁbered, there is a 3-connected graph Γ embedded in M such that TSG+(Γ, M) is not isomorphic to any ﬁnite group of homeomorphisms of M.

Erica Flapan Topological Symmetries of Molecules Thanks

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Erica Flapan Topological Symmetries of Molecules