two fold coverings and we’re not talking about plus size bikinis

Two Fold Coverings Two Fold Coverings and We’re Not Talking and We’re Not Talking About Plus Size BikinisAbout Plus Size Bikinis

Presenters:Presenters:Krista JoslinKrista Joslin

Carla RanalloCarla RanalloCylde TedrickCylde Tedrick

Steve EgleSteve EgleBrian KingBrian King

Mark HerriedMark HerriedNate ZimmerNate Zimmer

OutlineOutlineI) Topological StereochemistryI) Topological Stereochemistry

a) Molecular graphs as topological objects in spacea) Molecular graphs as topological objects in space

b) Topological Chirality and Achiralityb) Topological Chirality and Achirality

II) Molecular Moebius laddersII) Molecular Moebius ladders

a) Description and Backgrounda) Description and Background

b) Statement of Simon’s 1986 Theoremb) Statement of Simon’s 1986 Theorem

III) Topological Concepts and MachineryIII) Topological Concepts and Machinery

a) Topological Spacesa) Topological Spaces

b) Manifoldsb) Manifolds

c) Covering Spacesc) Covering Spaces

1) 2-fold Coverings1) 2-fold Coverings

2) 2-fold Branched Coverings2) 2-fold Branched Coverings

3) General Covering Spaces3) General Covering Spaces

IV) ConclusionIV) Conclusion

Topological Topological StereochemistryStereochemistry

Topological Stereochemistry Topological Stereochemistry is…is…

Stereochemistry is the study of stereoisomers Stereochemistry is the study of stereoisomers which are compounds that have the same which are compounds that have the same chemical formula and the same connectivity but chemical formula and the same connectivity but different arrangements of their atoms in a 3 – different arrangements of their atoms in a 3 – dimensional space.dimensional space.

Studies of synthesis, characterization and Studies of synthesis, characterization and analysis of molecular structures that are analysis of molecular structures that are topologically nontrivial.topologically nontrivial.

When can/cannot one embedded graph be When can/cannot one embedded graph be “deformed” into another?“deformed” into another?

What are the properties of embedded graphs that What are the properties of embedded graphs that are preserved by deformation?are preserved by deformation?

Graphical RepresentationGraphical Representation

A graph, G = (V, E), is a collection of vertices and A graph, G = (V, E), is a collection of vertices and edges, where V is the set of vertices and E the set edges, where V is the set of vertices and E the set of edges.of edges.

G is undirected if for two vertices, vG is undirected if for two vertices, vaa and v and vbb, the , the edge (vedge (vaa, v, vbb) is equal to the edge (v) is equal to the edge (vbb, v, vaa). ). Basically, all edges can flow in both directions.Basically, all edges can flow in both directions.

G is directed if for two vertices, vG is directed if for two vertices, vaa and v and vbb, ,

(v(vaa, v, vbb) ) (v (vbb, v, vaa).).v1

v3

v2

v4

Graphical Representation Graphical Representation (cont.)(cont.)

v1

v3

v2

v4

v1

v3

v2

v4

Undirected Graph Directed Graph

Two examples of graphical representations:Two examples of graphical representations:

** UndirectedUndirected

** DirectedDirected

AchiralityAchirality

Definition of AchiralityDefinition of Achirality

A graph embedded in RA graph embedded in R33 is topologically achiral if is topologically achiral if it can be deformed into its mirror image.it can be deformed into its mirror image.

** Deformation by a way of bending, twisting Deformation by a way of bending, twisting and/or and/or rotating without breaking or tearing rotating without breaking or tearing the molecule.the molecule.

Another way of observing achirality is Another way of observing achirality is symetrical symetrical elementselements. If the molecule or object has either a . If the molecule or object has either a plane of symmetry or a center of symmetry it is plane of symmetry or a center of symmetry it is achiral. achiral.

Example of AchiralityExample of Achirality

2 – proponol is an achiral organic molecule.2 – proponol is an achiral organic molecule.

Key:Key: BlueBlue is carbon is carbon

YellowYellow is CH3 Group is CH3 Group

RedRed is Oxygen is Oxygen

WhiteWhite is Hydrogen is Hydrogen

ChiralityChirality

Definition of ChiralityDefinition of Chirality

A graph embedded in RA graph embedded in R33 is topologically chiral if it is topologically chiral if it is not identical (i.e., non-superimposable upon) is not identical (i.e., non-superimposable upon) and and cannotcannot be deformed into its mirror image. be deformed into its mirror image.

** Once again defining deformation by a way Once again defining deformation by a way of bending, twisting and/or rotating without of bending, twisting and/or rotating without breaking or tearing the molecule.breaking or tearing the molecule.

Example of ChiralityExample of Chirality

Our hands are chiral, they cannot be deformed Our hands are chiral, they cannot be deformed into their mirror image.into their mirror image.

More examples of ChiralityMore examples of Chirality

Some examples of chirality in our world include…Some examples of chirality in our world include…

*Glucose (and all sugars)*Glucose (and all sugars)

*Proteins*Proteins

*Nucleic Acids*Nucleic Acids

*DNA*DNA

*As well as over half the organic compounds in *As well as over half the organic compounds in common drugscommon drugs

One more example of ChiralityOne more example of Chirality

2 – Butanol is a chiral molecule.2 – Butanol is a chiral molecule.

Key:Key: BlueBlue is carbon is carbon

YellowYellow is Methyl Group is Methyl Group

GreenGreen is Ethyl Group is Ethyl Group

RedRed is Oxygen is Oxygen

WhiteWhite is Hydrogen is Hydrogen

Mathematical Models Mathematical Models of Chiralityof Chirality

HomeomorphismHomeomorphism

Let Let hh: A : A B be a function. We say that B be a function. We say that hh is a is a homeomorphismhomeomorphism if if hh is continuous, and is continuous, and hh has a has a continuous inverse.continuous inverse.

Homeomorphisms are either differentiable or Homeomorphisms are either differentiable or piecewise linear.piecewise linear.

Ambient IsotopyAmbient Isotopy

Let A and B be contained in a set M , a Let A and B be contained in a set M , a mathematical model, which is a subset of Rmathematical model, which is a subset of Rnn. We . We say that A is ambient isotopic to B in M if there is say that A is ambient isotopic to B in M if there is a continuous function F:MxI a continuous function F:MxI M such that for M such that for each fixed t each fixed t I the function F(x,T) is a I the function F(x,T) is a homeomorphism, F(x,0) = x for all x homeomorphism, F(x,0) = x for all x M, and F(A M, and F(A x {1}) = B. The function F is said to be an x {1}) = B. The function F is said to be an Ambient IsotopyAmbient Isotopy..

We can now say…We can now say…

Achiral molecules are ambient Achiral molecules are ambient isotopic to their mirror imageisotopic to their mirror image

andand Chiral molecules are ambient isotopic Chiral molecules are ambient isotopic

to their mirror image to their mirror image

Topological Chirality/AchiralityTopological Chirality/Achirality

An embedded graph G An embedded graph G R R3 3 is topologically achiral is topologically achiral if there exists an orientation reversing if there exists an orientation reversing homeomorphism of (Rhomeomorphism of (R33, G). If not, G is , G). If not, G is topologically chiral.topologically chiral.

Two homeomorphisms are isotopic if Two homeomorphisms are isotopic if one can be continuously deformed one can be continuously deformed into the other.into the other.

An important thing to remember is An important thing to remember is that every homeomorphism is that every homeomorphism is isotopic to either the identity map or isotopic to either the identity map or to a reflection map, but not to both. to a reflection map, but not to both.

Two types of HomeomorphismsTwo types of Homeomorphisms

Let us consider h be a Let us consider h be a homeomorphism from Rhomeomorphism from R33 to itself… to itself…

If h is isotopic to the identity map, then If h is isotopic to the identity map, then we say that h is we say that h is orientation preserving.orientation preserving.

If h is isotopic to a reflection map, then If h is isotopic to a reflection map, then we say that h is we say that h is orientation reversing. orientation reversing.

Mobius LadderMobius Ladder

A mobius ladder, MA mobius ladder, Mn n consists of a simple consists of a simple closed curve K with 2n vertices. closed curve K with 2n vertices. Together with n additional edges a1,Together with n additional edges a1,….,an such that if the vertices on the ….,an such that if the vertices on the curve K are consecutively labeled 1,2,3,curve K are consecutively labeled 1,2,3,…..,2n then the vertices of each edge a …..,2n then the vertices of each edge a then the vertices of each edge athen the vertices of each edge aii are I are I and I + n. K is the loop of the mobius and I + n. K is the loop of the mobius ladder Mladder Mn n and a,…,an are the rungs of Mand a,…,an are the rungs of Mn.n.

Mobius Ladder chiralityMobius Ladder chirality

Catenane (# 467)Catenane (# 467)

Left and right handed Mobius laddersLeft and right handed Mobius ladders

Applications of CatenaneApplications of Catenane

Molecular Memory for Computers Molecular Memory for Computers

Molecular memoryMolecular memory

Random access data storage could be Random access data storage could be provided by rings of atoms. Researchers provided by rings of atoms. Researchers who have developed a system of who have developed a system of microscopic chemical switches that could microscopic chemical switches that could form the basis of tiny, fast and cheap form the basis of tiny, fast and cheap computers. This system could allow our computers. This system could allow our computers to do things that we cannot computers to do things that we cannot even imagine now.even imagine now.

How Molecular memory works?How Molecular memory works?

A pulse of electricity would remove one A pulse of electricity would remove one electron, thus causing one ring to flip or electron, thus causing one ring to flip or rotate around the other. This is how the rotate around the other. This is how the switch would be turned on. Putting an switch would be turned on. Putting an electron back turns the switch off.electron back turns the switch off.

Works at room temperature.Works at room temperature.

MolecularMolecularMemoryMemory

It is also easy to see whether or not the It is also easy to see whether or not the catenane is working. "It is green in the catenane is working. "It is green in the starting state ... and then it switches to starting state ... and then it switches to being maroon, you can use your eyes to being maroon, you can use your eyes to detect it.detect it.

Jon Simon’s Theorem (1986)Jon Simon’s Theorem (1986)

Proved that embedded graphs Proved that embedded graphs representing the molecular Mobius representing the molecular Mobius ladders with an odd number or rungs ladders with an odd number or rungs greater than two is necessarily greater than two is necessarily topologically chiral.topologically chiral.

In contrast, a Mobius ladder with an In contrast, a Mobius ladder with an even number or rungs has a even number or rungs has a topologically achiral embedding.topologically achiral embedding.

Jon Simon’s Theorem (1986) Jon Simon’s Theorem (1986)

Used topological machinery to prove Used topological machinery to prove his theory. Such topological his theory. Such topological concepts and machinery used were concepts and machinery used were topological spaces and covering topological spaces and covering spaces.spaces.

Topological Topological ConceptsConcepts

and Machineryand Machinery

Topological SpacesTopological Spaces

Topological SpacesTopological Spaces

A topology on a set X is a collection A topology on a set X is a collection ΤΤ if subsets of X have these properties:if subsets of X have these properties:

i) i) Ǿ, X € Ǿ, X € ΤΤ ii) The union of the elements of any ii) The union of the elements of any

subcollection of subcollection of ΤΤ is inis in Τ Τ iii) The intersection of the elements iii) The intersection of the elements

of any finite subcollection of of any finite subcollection of ΤΤ is in is in Τ Τ

Topological Spaces ContinuedTopological Spaces Continued

A set X, for which a topology A set X, for which a topology ΤΤ has has been specified is called a been specified is called a topological space topological space written (X,T)written (X,T)

If we have a topological space (X,T) If we have a topological space (X,T) and Uand UcX, U€T, U is called an “open cX, U€T, U is called an “open set”set”


We can say a topological space is a We can say a topological space is a set X together with a collection of set X together with a collection of subsets of X, called open sets such subsets of X, called open sets such that that Ǿ, X are both open, arbitrary Ǿ, X are both open, arbitrary unions of open sets are open and unions of open sets are open and finite intersections of open sets are finite intersections of open sets are openopen


Ex) X Ex) X ~ a set, ~ a set, T T = all subsets of = all subsets of X, this is called the discrete X, this is called the discrete topologytopology

Ex) If {X,Ǿ} = Ex) If {X,Ǿ} = TT, this is called , this is called the trivial or indiscrete the trivial or indiscrete topologytopology

Ex) Let X= (a,b,c} then the Ex) Let X= (a,b,c} then the discrete topology discrete topology TT ={ X, Ǿ, ={ X, Ǿ, {a}, {b}, {c}, {a,b}, {a,c}, {a}, {b}, {c}, {a,b}, {a,c}, {b,c} } {b,c} }


BasisBasis Let (X, Let (X, TT) be a topological space. ) be a topological space.

A basis B is a collection of A basis B is a collection of subsets X, (called basis subsets X, (called basis elements) such that:elements) such that:

(1) (1) x xX X B B B, X B, X B B (2) if X (2) if X B B1 1 B B2 2 (B(B11,B,B2 2 B) then B) then

BB3 3 B B X X BB3 3 B B1 1 BB22

ManifoldsManifolds

A Brief IntroductionA Brief Introduction

In essence, a In essence, a manifoldmanifold is a space that is a space that is locally like Ris locally like Rnn, however lacking a , however lacking a preferred system of coordinates. preferred system of coordinates. Furthermore, a manifold can have Furthermore, a manifold can have global topological properties that global topological properties that distinguish it from the topologically distinguish it from the topologically trivial Rtrivial Rnn..

DefinitionDefinition

Let M be a subset of RLet M be a subset of Rpp for some p. A for some p. A subset U of M is said to be open in M subset U of M is said to be open in M if U equals the intersection of M and if U equals the intersection of M and V where V is an open set in RV where V is an open set in Rpp. Let n . Let n be a natural number. We say that M be a natural number. We say that M is an n-manifold if each point x of M is an n-manifold if each point x of M is contained in an open set U of M is contained in an open set U of M that is either homeomorphic to Rthat is either homeomorphic to Rnn or or to the half-space Rto the half-space R+n+n..

p does not always imply np does not always imply n

For an example think of a cover of a For an example think of a cover of a baseball as if it were a hollow sphere, baseball as if it were a hollow sphere, which would be an example of a two-which would be an example of a two-manifold, while the rubber or cork manifold, while the rubber or cork ball and the twine that makes up the ball and the twine that makes up the solid center would be an example of solid center would be an example of a three-manifold. Both of these a three-manifold. Both of these manifolds are subsets of 3 - manifolds are subsets of 3 - dimesional space (Rdimesional space (R33). ).

The coverThe cover

We can imagine a baseball imbedded in We can imagine a baseball imbedded in three sphere, and let T denote the three sphere, and let T denote the surface area of the baseball. For any surface area of the baseball. For any point x contained within T, we can point x contained within T, we can choose V (similar to epsilon choose V (similar to epsilon neighborhoods in Real Analysis) to be a neighborhoods in Real Analysis) to be a small open ball (in Rsmall open ball (in R33) whose center ) whose center falls on x. By definition since V is open falls on x. By definition since V is open (in R(in R33), the set U which is given by the ), the set U which is given by the intersection of sets T and V is also intersection of sets T and V is also open. open.

The cover (continued)The cover (continued)

If you choose a small enough radius If you choose a small enough radius for V, then the resulting U will yield a for V, then the resulting U will yield a small slightly curved disk (similar to small slightly curved disk (similar to a plumping piece of pepperoni on a a plumping piece of pepperoni on a cooking pizza) whose interior is cooking pizza) whose interior is homeomorphic to the interior of a flat homeomorphic to the interior of a flat disk. A flat disk in turn is disk. A flat disk in turn is homeomorphic to Rhomeomorphic to R22. .

Ball and twineBall and twine

We can similarly argue that a solid We can similarly argue that a solid sphere (denoted by S) is a three-sphere (denoted by S) is a three-manifold. Only in this case the manifold. Only in this case the interior points of S are contained in interior points of S are contained in an open set that is homeomorphic to an open set that is homeomorphic to RR33..

So what are some examples of So what are some examples of manifolds?manifolds?

One-manifolds: line segments, lines, One-manifolds: line segments, lines, circles, and unions of thesecircles, and unions of these

Two-manifolds (surfaces): Mobius Two-manifolds (surfaces): Mobius strip, annulus, the surface of a strip, annulus, the surface of a sphere.sphere.

Three-manifolds: a three dimesional Three-manifolds: a three dimesional sphere, three dimensional ball, torus sphere, three dimensional ball, torus (doughnut)(doughnut)

Why the interest in manifolds?Why the interest in manifolds?

In general, manifolds have generated In general, manifolds have generated so much interest because they are so much interest because they are easier to deal with than other easier to deal with than other subsets of Rsubsets of Rnn. .

Covering Covering SpacesSpaces(2-fold coverings)(2-fold coverings)

Open sets and N-ManifoldsOpen sets and N-Manifolds

Let M be a subset of RLet M be a subset of Rpp for some p. A for some p. A subset U of M is said to be open in M subset U of M is said to be open in M if U=Mif U=MV where V is an open set in V where V is an open set in RRpp. Let n be a natural number. We . Let n be a natural number. We say that M is an n-manifold if each say that M is an n-manifold if each point x of M is contained in an open point x of M is contained in an open set U of M that is either set U of M that is either homeomorphic to Rhomeomorphic to Rnn or to the half- or to the half-space Rspace Rnn+ = {(x+ = {(x11,…,x,…,xnn) ) R Rnn | x | xnn > 0}. > 0}.

Order of a homeomorphismOrder of a homeomorphism

Def:Def: Let M be a subset of R Let M be a subset of Rnn and let and let h:Mh:MM be a homeomorphism. Let r be M be a homeomorphism. Let r be an natural number. Then han natural number. Then hrr is the is the homeomorphism is obtained by homeomorphism is obtained by performing h some number r times. If performing h some number r times. If r is the smallest number such that hr is r is the smallest number such that hr is the identity map, then we say h as an the identity map, then we say h as an order of r. If there is no such r then we order of r. If there is no such r then we say h does not have finite order.say h does not have finite order.

Covering InvolutionCovering Involution

Let h:MLet h:MM be an orientation M be an orientation preserving homeomorphism of order preserving homeomorphism of order two.two.

Projection MapProjection Map

Def: Let M and N be three manifolds. Def: Let M and N be three manifolds. Let p:MLet p:MN be a function, that is N be a function, that is continuous and takes open sets to continuous and takes open sets to open sets. If p(x)=p(y) if and only if open sets. If p(x)=p(y) if and only if either x=y or h(x)=y the p is said to either x=y or h(x)=y the p is said to be a projection map.be a projection map.

Twofold CoverTwofold Cover

Let M and N be three manifolds, and let Let M and N be three manifolds, and let h:Mh:MM be a covering involution. Let M be a covering involution. Let p:Mp:MN be a projection map. Let A denote N be a projection map. Let A denote the set of points x in M such that h(x)=x. the set of points x in M such that h(x)=x.

If B=p(A) is a one-manifold then we say M If B=p(A) is a one-manifold then we say M is a twofold branch cover of N branched is a twofold branch cover of N branched over B. If A is the empty set then we say over B. If A is the empty set then we say M is a twofold cover of N.M is a twofold cover of N.

Twofold Branch Twofold Branch CoveringCovering

Twofold Branch Cover Twofold Branch Cover ExplanationExplanation

Consider the three-manifolds M and N.Consider the three-manifolds M and N. Since h: M -> M it just transfers between the Since h: M -> M it just transfers between the

different layers of M.different layers of M. Twofold Branch Covers means that h(x) = x, so h Twofold Branch Covers means that h(x) = x, so h

moves along a fixed point on each disk.moves along a fixed point on each disk. The function p(x):M -> N will transfer you from The function p(x):M -> N will transfer you from

the disks M to the disk N by wrapping M around N the disks M to the disk N by wrapping M around N twice, hence the name Twofold branch cover.twice, hence the name Twofold branch cover.

h(x)

p(x)

M N

2-D Example of a Twofold 2-D Example of a Twofold Branched CoverBranched Cover

Let Let MM denote a unit disk expressed in polar denote a unit disk expressed in polar coordinatescoordinates

Define Define h: M -> Mh: M -> M by by h(r,h(r,) = (r, ) = (r, so so that that hh rotates rotates MM by by

Define Define p: M -> Mp: M -> M by by p(r, p(r, ) = (r,2) = (r,2)), wrapping , wrapping MM around itself twice around itself twice

Functions Functions hh and and pp are related because for every are related because for every xx,, y € M y € M, we have , we have p(x) = p(y)p(x) = p(y) if and only if if and only if either either x = yx = y or or h(x) = yh(x) = y

(r,(r,

h

2-D Example (cont.)2-D Example (cont.) Let Let MM11 be the surface obtained by cutting be the surface obtained by cutting

MM open along a single radius of the disk open along a single radius of the disk Let Let MM22 be a copy of be a copy of MM11

Stretch Stretch MM11 and and MM22 open, and disk open, and disk MM is is obtained by gluing these two half-disks obtained by gluing these two half-disks togethertogether

Therefore Therefore hh interchanges interchanges MM11 and and MM22, and , and pp sends each of sends each of MM11 and and MM22 onto onto MM

M1 M2 M

2-D Example (cont.)2-D Example (cont.)

This example has This example has hh where it fixes the where it fixes the center point of center point of MM

Therefore Therefore MM is a twofold branched is a twofold branched cover of itself with branch set the cover of itself with branch set the center point of center point of MM

h(x)

M1

M2

p(x) M

General Covering General Covering SpacesSpaces

Def: Let P:EDef: Let P:E B (where P is the B (where P is the projection map) be continuous and projection map) be continuous and surjective. If every point b of B has a surjective. If every point b of B has a neighborhood U that is evenly neighborhood U that is evenly covered by Pcovered by P1 1 the P is called a the P is called a covering map and E is said to be a covering map and E is said to be a covering space of B.covering space of B.

Projection (P) of E onto B Projection (P) of E onto B

E

B

P

U

Covering InvolutionCovering Involution H:EH:E F F jumping between “pancakes” jumping between “pancakes”

Example of covering Involution is like Example of covering Involution is like the mapping of a number line onto a the mapping of a number line onto a circle. Instead mapping a number circle. Instead mapping a number line onto itself around a circle, line onto itself around a circle, these are all these are all mappings of the same point on the mappings of the same point on the number unit circle.number unit circle.

ConclusionConclusion Ambient isotopy and homeomorphisms are Ambient isotopy and homeomorphisms are

graphical representations of chirality/achiralitygraphical representations of chirality/achirality Achirality/chirality are important concepts to Achirality/chirality are important concepts to

the fields of chemistry and biology the fields of chemistry and biology With machinery gained from Jon Simon’s proof With machinery gained from Jon Simon’s proof

such as:such as: Topological spaces, manifolds, and coverings Topological spaces, manifolds, and coverings

spacesspaces we gain new knowledge on how to solve we gain new knowledge on how to solve

present problems today, especially dealing present problems today, especially dealing with nano-technology.with nano-technology.

SourcesSources

www.cosmiverse.comwww.cosmiverse.com ““When Chemistry Meets Topology” by When Chemistry Meets Topology” by

Erica FlapanErica Flapan ““Topological Chirality of Certain Topological Chirality of Certain

Molecules” by Jonathon SimonMolecules” by Jonathon Simon ““Molecular Graphs as Topological Molecular Graphs as Topological

Objects in Space” by Jonathon SimonObjects in Space” by Jonathon Simon Professor Steve DeckelmanProfessor Steve Deckelman

two fold coverings and we’re not talking about plus size bikinis

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