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

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  • Slide 1
  • Slide 2
  • Two Fold Coverings and Were Not Talking About Plus Size Bikinis
  • Slide 3
  • Presenters: Krista Joslin Carla Ranallo Cylde Tedrick Steve Egle Brian King Mark Herried Nate Zimmer
  • Slide 4
  • Outline I) Topological Stereochemistry a) Molecular graphs as topological objects in space a) Molecular graphs as topological objects in space b) Topological Chirality and Achirality b) Topological Chirality and Achirality II) Molecular Moebius ladders a) Description and Background a) Description and Background b) Statement of Simons 1986 Theorem b) Statement of Simons 1986 Theorem III) Topological Concepts and Machinery a) Topological Spaces a) Topological Spaces b) Manifolds b) Manifolds c) Covering Spaces c) Covering Spaces 1) 2-fold Coverings 1) 2-fold Coverings 2) 2-fold Branched Coverings 3) General Covering Spaces IV) Conclusion
  • Slide 5
  • Topological Stereochemistry
  • Slide 6
  • Topological Stereochemistry is Stereochemistry is the study of stereoisomers which are compounds that have the same chemical formula and the same connectivity but different arrangements of their atoms in a 3 dimensional space. Stereochemistry is the study of stereoisomers which are compounds that have the same chemical formula and the same connectivity but different arrangements of their atoms in a 3 dimensional space. Studies of synthesis, characterization and analysis of molecular structures that are topologically nontrivial. Studies of synthesis, characterization and analysis of molecular structures that are topologically nontrivial. When can/cannot one embedded graph be deformed into another? When can/cannot one embedded graph be deformed into another? What are the properties of embedded graphs that are preserved by deformation? What are the properties of embedded graphs that are preserved by deformation?
  • Slide 7
  • Graphical Representation A graph, G = (V, E), is a collection of vertices and edges, where V is the set of vertices and E the set of edges. A graph, G = (V, E), is a collection of vertices and edges, where V is the set of vertices and E the set of edges. G is undirected if for two vertices, v a and v b, the edge (v a, v b ) is equal to the edge (v b, v a ). Basically, all edges can flow in both directions. G is undirected if for two vertices, v a and v b, the edge (v a, v b ) is equal to the edge (v b, v a ). Basically, all edges can flow in both directions. G is directed if for two vertices, v a and v b, G is directed if for two vertices, v a and v b, (v a, v b ) (v b, v a ). v1v1 v3v3 v2v2 v4v4
  • Slide 8
  • Graphical Representation (cont.) v1v1 v3v3 v2v2 v4v4 v1v1 v3v3 v2v2 v4v4 Undirected GraphDirected Graph Two examples of graphical representations: Two examples of graphical representations: *Undirected *Directed
  • Slide 9
  • Achirality
  • Slide 10
  • Definition of Achirality A graph embedded in R 3 is topologically achiral if it can be deformed into its mirror image. A graph embedded in R 3 is topologically achiral if it can be deformed into its mirror image. *Deformation by a way of bending, twisting and/or rotating without breaking or tearing the molecule. Another way of observing achirality is symetrical elements. If the molecule or object has either a plane of symmetry or a center of symmetry it is achiral. Another way of observing achirality is symetrical elements. If the molecule or object has either a plane of symmetry or a center of symmetry it is achiral.
  • Slide 11
  • Example of Achirality 2 proponol is an achiral organic molecule. 2 proponol is an achiral organic molecule. Key:Blue is carbon Yellow is CH3 Group Red is Oxygen White is Hydrogen
  • Slide 12
  • Chirality
  • Slide 13
  • Definition of Chirality A graph embedded in R 3 is topologically chiral if it is not identical (i.e., non-superimposable upon) and cannot be deformed into its mirror image. A graph embedded in R 3 is topologically chiral if it is not identical (i.e., non-superimposable upon) and cannot be deformed into its mirror image. *Once again defining deformation by a way of bending, twisting and/or rotating without breaking or tearing the molecule. *Once again defining deformation by a way of bending, twisting and/or rotating without breaking or tearing the molecule.
  • Slide 14
  • Example of Chirality Our hands are chiral, they cannot be deformed into their mirror image. Our hands are chiral, they cannot be deformed into their mirror image.
  • Slide 15
  • More examples of Chirality Some examples of chirality in our world include Some examples of chirality in our world include *Glucose (and all sugars) *Proteins *Nucleic Acids *DNA *As well as over half the organic compounds in common drugs
  • Slide 16
  • One more example of Chirality 2 Butanol is a chiral molecule. 2 Butanol is a chiral molecule. Key:Blue is carbon Yellow is Methyl Group Green is Ethyl Group Red is Oxygen White is Hydrogen
  • Slide 17
  • Mathematical Models of Chirality
  • Slide 18
  • Homeomorphism Let h: A B be a function. We say that h is a homeomorphism if h is continuous, and h has a continuous inverse. Let h: A B be a function. We say that h is a homeomorphism if h is continuous, and h has a continuous inverse. Homeomorphisms are either differentiable or piecewise linear. Homeomorphisms are either differentiable or piecewise linear.
  • Slide 19
  • Ambient Isotopy Let A and B be contained in a set M, a mathematical model, which is a subset of R n. We say that A is ambient isotopic to B in M if there is a continuous function F:MxI M such that for each fixed t I the function F(x,T) is a homeomorphism, F(x,0) = x for all x M, and F(A x {1}) = B. The function F is said to be an Ambient Isotopy. Let A and B be contained in a set M, a mathematical model, which is a subset of R n. We say that A is ambient isotopic to B in M if there is a continuous function F:MxI M such that for each fixed t I the function F(x,T) is a homeomorphism, F(x,0) = x for all x M, and F(A x {1}) = B. The function F is said to be an Ambient Isotopy.
  • Slide 20
  • We can now say Achiral molecules are ambient isotopic to their mirror image Achiral molecules are ambient isotopic to their mirror imageand Chiral molecules are ambient isotopic to their mirror image Chiral molecules are ambient isotopic to their mirror image
  • Slide 21
  • Topological Chirality/Achirality An embedded graph G R 3 is topologically achiral if there exists an orientation reversing homeomorphism of (R 3, G). If not, G is topologically chiral. An embedded graph G R 3 is topologically achiral if there exists an orientation reversing homeomorphism of (R 3, G). If not, G is topologically chiral.
  • Slide 22
  • Two homeomorphisms are isotopic if one can be continuously deformed into the other. Two homeomorphisms are isotopic if one can be continuously deformed into the other. An important thing to remember is that every homeomorphism is isotopic to either the identity map or to a reflection map, but not to both. An important thing to remember is that every homeomorphism is isotopic to either the identity map or to a reflection map, but not to both.
  • Slide 23
  • Two types of Homeomorphisms Let us consider h be a homeomorphism from R 3 to itself Let us consider h be a homeomorphism from R 3 to itself If h is isotopic to the identity map, then we say that h is orientation preserving. If h is isotopic to the identity map, then we say that h is orientation preserving. If h is isotopic to a reflection map, then we say that h is orientation reversing. If h is isotopic to a reflection map, then we say that h is orientation reversing.
  • Slide 24
  • Mobius Ladder A mobius ladder, M n consists of a simple closed curve K with 2n vertices. Together with n additional edges a1,.,an such that if the vertices on the curve K are consecutively labeled 1,2,3,..,2n then the vertices of each edge a then the vertices of each edge a i are I and I + n. K is the loop of the mobius ladder M n and a,,an are the rungs of M n. A mobius ladder, M n consists of a simple closed curve K with 2n vertices. Together with n additional edges a1,.,an such that if the vertices on the curve K are consecutively labeled 1,2,3,..,2n then the vertices of each edge a then the vertices of each edge a i are I and I + n. K is the loop of the mobius ladder M n and a,,an are the rungs of M n.
  • Slide 25
  • Mobius Ladder chirality
  • Slide 26
  • Catenane (# 467) Left and right handed Mobius ladders Left and right handed Mobius ladders
  • Slide 27
  • Applications of Catenane Molecular Memory for Computers Molecular Memory for Computers
  • Slide 28
  • Molecular memory Random access data storage could be provided by rings of atoms. Researchers who have developed a system of microscopic chemical switches that could form the basis of tiny, fast and cheap computers. This system could allow our computers to do things that we cannot even imagine now. Random access data storage could be provided by rings of atoms. Researchers who have developed a system of microscopic chemical switches that could form the basis of tiny, fast and cheap computers. This system could allow our computers to do things that we cannot even imagine now.
  • Slide 29
  • How Molecular memory works? A pulse of electricity would remove one electron, thus causing one ring to flip or rotate around the other. This is how the switch would be turned on. Putting an electron back turns the switch off. A pulse of electricity would remove one electron, thus causing one ring to flip or rotate around the other. This is how the switch would be turned on. Putting an electron back turns the switch off. Works at room temperature. Works at room temperature.
  • Slide 30
  • Molecular Memory It is also easy to see whether or not the catenane is working. "It is green in the starting state... and then it switches to being maroon, you can use your eyes to detect it. It is also easy to see whether or not the catenane is working. "It is green in the starting state... and then it switches to being maroon, you can use your eyes to detect it.
  • Slide 31
  • Jon Simons Theorem (1986) Proved that embedded graphs representing the molecular Mobius ladders with an odd number or rungs greater than two is necessarily topologically chiral. Proved that embedded graphs representing the molecular Mobius ladders with an odd number or rungs greater than two is necessarily topologically chiral. In contrast, a Mobius ladder with an even number or rungs has a topologically achiral embedding. In contrast, a Mobius ladder with an even number or rungs has a topologically achiral embedding.
  • Slide 32
  • Jon Simons Theorem (1986) Used topological machinery to prove his theory. Such topological concepts and machinery used were topological spaces and covering spaces. Used topological machinery to prove his theory. Such topological concepts and machinery used were topological spaces and covering spaces.
  • Slide 33
  • Topological Concepts and Machinery
  • Slide 34
  • Topological Spaces
  • Slide 35
  • A topology on a set X is a collection if subsets of X have these properties: A topology on a set X is a collection if subsets of X have these properties: i) , X i) , X ii) The union of the elements of any subcollection of is in ii) The union of the elements of any subcollection of is in iii) The intersection of the elements of any finite subcollection of is in iii) The intersection of the elements of any finite subcollection of is in
  • Slide 36
  • Topological Spaces Continued A set X, for which a topology has been specified is called a topological space written (X,T) A set X, for which a topology has been specified is called a topological space written (X,T) If we have a topological space (X,T) and UcX, UT, U is called an open set If we have a topological space (X,T) and UcX, UT, U is called an open set
  • Slide 37
  • Topological Spaces Continued We can say a topological space is a set X together with a collection of subsets of X, called open sets such that , X are both open, arbitrary unions of open sets are open and finite intersections of open sets are open We can say a topological space is a set X together with a collection of subsets of X, called open sets such that , X are both open, arbitrary unions of open sets are open and finite intersections of open sets are open
  • Slide 38
  • Topological Spaces Continued Ex) X ~ a set, T = all subsets of X, this is called the discrete topology Ex) X ~ a set, T = all subsets of X, this is called the discrete topology Ex) If {X,} = T, this is called the trivial or indiscrete topology Ex) If {X,} = T, this is called the trivial or indiscrete topology Ex) Let X= (a,b,c} then the discrete topology T ={ X, , {a}, {b}, {c}, {a,b}, {a,c}, {b,c} } Ex) Let X= (a,b,c} then the discrete topology T ={ X, , {a}, {b}, {c}, {a,b}, {a,c}, {b,c} }
  • Slide 39
  • Topological Spaces Continued Basis Basis Let (X, T ) be a topological space. A basis B is a collection of subsets X, (called basis elements) such that: Let (X, T ) be a topological space. A basis B is a collection of subsets X, (called basis elements) such that: (1) x X B B, X B (1) x X B B, X B (2) if X B 1 B 2 (B 1,B 2 B) then B 3 B X B 3 B 1 B 2 (2) if X B 1 B 2 (B 1,B 2 B) then B 3 B X B 3 B 1 B 2
  • Slide 40
  • Manifolds
  • Slide 41
  • A Brief Introduction In essence, a manifold is a space that is locally like R n, however lacking a preferred system of coordinates. Furthermore, a manifold can have global topological properties that distinguish it from the topologically trivial R n. In essence, a manifold is a space that is locally like R n, however lacking a preferred system of coordinates. Furthermore, a manifold can have global topological properties that distinguish it from the topologically trivial R n.
  • Slide 42
  • Definition Let M be a subset of R p for some p. A subset U of M is said to be open in M if U equals the intersection of M and V where V is an open set in R p. Let n be a natural number. We say that M is an n-manifold if each point x of M is contained in an open set U of M that is either homeomorphic to R n or to the half-space R +n. Let M be a subset of R p for some p. A subset U of M is said to be open in M if U equals the intersection of M and V where V is an open set in R p. Let n be a natural number. We say that M is an n-manifold if each point x of M is contained in an open set U of M that is either homeomorphic to R n or to the half-space R +n.
  • Slide 43
  • p does not always imply n For an example think of a cover of a baseball as if it were a hollow sphere, which would be an example of a two- manifold, while the rubber or cork ball and the twine that makes up the solid center would be an example of a three-manifold. Both of these manifolds are subsets of 3 - dimesional space (R 3 ). For an example think of a cover of a baseball as if it were a hollow sphere, which would be an example of a two- manifold, while the rubber or cork ball and the twine that makes up the solid center would be an example of a three-manifold. Both of these manifolds are subsets of 3 - dimesional space (R 3 ).
  • Slide 44
  • The cover We can imagine a baseball imbedded in three sphere, and let T denote the surface area of the baseball. For any point x contained within T, we can choose V (similar to epsilon neighborhoods in Real Analysis) to be a small open ball (in R 3 ) whose center falls on x. By definition since V is open (in R 3 ), the set U which is given by the intersection of sets T and V is also open. We can imagine a baseball imbedded in three sphere, and let T denote the surface area of the baseball. For any point x contained within T, we can choose V (similar to epsilon neighborhoods in Real Analysis) to be a small open ball (in R 3 ) whose center falls on x. By definition since V is open (in R 3 ), the set U which is given by the intersection of sets T and V is also open.
  • Slide 45
  • The cover (continued) If you choose a small enough radius for V, then the resulting U will yield a small slightly curved disk (similar to a plumping piece of pepperoni on a cooking pizza) whose interior is homeomorphic to the interior of a flat disk. A flat disk in turn is homeomorphic to R 2. If you choose a small enough radius for V, then the resulting U will yield a small slightly curved disk (similar to a plumping piece of pepperoni on a cooking pizza) whose interior is homeomorphic to the interior of a flat disk. A flat disk in turn is homeomorphic to R 2.
  • Slide 46
  • Ball and twine We can similarly argue that a solid sphere (denoted by S) is a three-manifold. Only in this case the interior points of S are contained in an open set that is homeomorphic to R 3. We can similarly argue that a solid sphere (denoted by S) is a three-manifold. Only in this case the interior points of S are contained in an open set that is homeomorphic to R 3.
  • Slide 47
  • So what are some examples of manifolds? One-manifolds: line segments, lines, circles, and unions of these One-manifolds: line segments, lines, circles, and unions of these
  • Slide 48
  • Two-manifolds (surfaces): Mobius strip, annulus, the surface of a sphere. Two-manifolds (surfaces): Mobius strip, annulus, the surface of a sphere.
  • Slide 49
  • Three-manifolds: a three dimesional sphere, three dimensional ball, torus (doughnut) Three-manifolds: a three dimesional sphere, three dimensional ball, torus (doughnut)
  • Slide 50
  • Why the interest in manifolds? In general, manifolds have generated so much interest because they are easier to deal with than other subsets of R n. In general, manifolds have generated so much interest because they are easier to deal with than other subsets of R n.
  • Slide 51
  • Covering Spaces (2-fold coverings)
  • Slide 52
  • Open sets and N-Manifolds Let M be a subset of R p for some p. A subset U of M is said to be open in M if U=M V where V is an open set in R p. Let n be a natural number. We say that M is an n-manifold if each point x of M is contained in an open set U of M that is either homeomorphic to R n or to the half- space R n + = {(x 1,,x n ) R n | x n > 0}. Let M be a subset of R p for some p. A subset U of M is said to be open in M if U=M V where V is an open set in R p. Let n be a natural number. We say that M is an n-manifold if each point x of M is contained in an open set U of M that is either homeomorphic to R n or to the half- space R n + = {(x 1,,x n ) R n | x n > 0}.
  • Slide 53
  • Order of a homeomorphism Def: Let M be a subset of R n and let h:M M be a homeomorphism. Let r be an natural number. Then h r is the homeomorphism is obtained by performing h some number r times. If r is the smallest number such that hr is the identity map, then we say h as an order of r. If there is no such r then we say h does not have finite order.
  • Slide 54
  • Covering Involution Let h:M M be an orientation preserving homeomorphism of order two. Let h:M M be an orientation preserving homeomorphism of order two.
  • Slide 55
  • Projection Map Def: Let M and N be three manifolds. Let p:M N be a function, that is continuous and takes open sets to open sets. If p(x)=p(y) if and only if either x=y or h(x)=y the p is said to be a projection map. Def: Let M and N be three manifolds. Let p:M N be a function, that is continuous and takes open sets to open sets. If p(x)=p(y) if and only if either x=y or h(x)=y the p is said to be a projection map.
  • Slide 56
  • Twofold Cover Let M and N be three manifolds, and let h:M M be a covering involution. Let p:M N be a projection map. Let A denote the set of points x in M such that h(x)=x. Let M and N be three manifolds, and let h:M M be a covering involution. Let p:M N be a projection map. Let A denote the set of points x in M such that h(x)=x. If B=p(A) is a one-manifold then we say M is a twofold branch cover of N branched over B. If A is the empty set then we say M is a twofold cover of N. If B=p(A) is a one-manifold then we say M is a twofold branch cover of N branched over B. If A is the empty set then we say M is a twofold cover of N.
  • Slide 57
  • Twofold Branch Covering
  • Slide 58
  • Twofold Branch Cover Explanation Consider the three-manifolds M and N. Consider the three-manifolds M and N. Since h: M -> M it just transfers between the different layers of M. Since h: M -> M it just transfers between the different layers of M. Twofold Branch Covers means that h(x) = x, so h moves along a fixed point on each disk. Twofold Branch Covers means that h(x) = x, so h moves along a fixed point on each disk. The function p(x):M -> N will transfer you from the disks M to the disk N by wrapping M around N twice, hence the name Twofold branch cover. The function p(x):M -> N will transfer you from the disks M to the disk N by wrapping M around N twice, hence the name Twofold branch cover. h(x) p(x) MN
  • Slide 59
  • 2-D Example of a Twofold Branched Cover Let M denote a unit disk expressed in polar coordinates Let M denote a unit disk expressed in polar coordinates Define h: M -> M by h(r, ) = (r, so that h rotates M by Define h: M -> M by h(r, ) = (r, so that h rotates M by Define p: M -> M by p(r, ) = (r,2 ), wrapping M around itself twice Define p: M -> M by p(r, ) = (r,2 ), wrapping M around itself twice Functions h and p are related because for every x, y M, we have p(x) = p(y) if and only if either x = y or h(x) = y Functions h and p are related because for every x, y M, we have p(x) = p(y) if and only if either x = y or h(x) = y (r, (r, h
  • Slide 60
  • 2-D Example (cont.) Let M 1 be the surface obtained by cutting M open along a single radius of the disk Let M 1 be the surface obtained by cutting M open along a single radius of the disk Let M 2 be a copy of M 1 Let M 2 be a copy of M 1 Stretch M 1 and M 2 open, and disk M is obtained by gluing these two half-disks together Stretch M 1 and M 2 open, and disk M is obtained by gluing these two half-disks together Therefore h interchanges M 1 and M 2, and p sends each of M 1 and M 2 onto M Therefore h interchanges M 1 and M 2, and p sends each of M 1 and M 2 onto M M1M1 M2M2 M
  • Slide 61
  • 2-D Example (cont.) This example has h where it fixes the center point of M This example has h where it fixes the center point of M Therefore M is a twofold branched cover of itself with branch set the center point of M Therefore M is a twofold branched cover of itself with branch set the center point of M h(x) M1M1 M2M2 p(x) M
  • Slide 62
  • General Covering Spaces
  • Slide 63
  • Def: Let P:E B (where P is the projection map) be continuous and surjective. If every point b of B has a neighborhood U that is evenly covered by P 1 the P is called a covering map and E is said to be a covering space of B. Def: Let P:E B (where P is the projection map) be continuous and surjective. If every point b of B has a neighborhood U that is evenly covered by P 1 the P is called a covering map and E is said to be a covering space of B.
  • Slide 64
  • Projection (P) of E onto B Projection (P) of E onto B E B P U
  • Slide 65
  • Covering Involution Covering Involution H:E F H:E F jumping between pancakes jumping between pancakes
  • Slide 66
  • Example of covering Involution is like the mapping of a number line onto a circle. Instead mapping a number line onto itself around a circle, these are all mappings of the same point on the number unit circle. Example of covering Involution is like the mapping of a number line onto a circle. Instead mapping a number line onto itself around a circle, these are all mappings of the same point on the number unit circle.
  • Slide 67
  • Conclusion Ambient isotopy and homeomorphisms are graphical representations of chirality/achirality Ambient isotopy and homeomorphisms are graphical representations of chirality/achirality Achirality/chirality are important concepts to the fields of chemistry and biology Achirality/chirality are important concepts to the fields of chemistry and biology With machinery gained from Jon Simons proof such as: With machinery gained from Jon Simons proof such as: Topological spaces, manifolds, and coverings spaces Topological spaces, manifolds, and coverings spaces we gain new knowledge on how to solve present problems today, especially dealing with nano- technology. we gain new knowledge on how to solve present problems today, especially dealing with nano- technology.
  • Slide 68
  • Sources www.cosmiverse.com www.cosmiverse.com When Chemistry Meets Topology by Erica Flapan When Chemistry Meets Topology by Erica Flapan Topological Chirality of Certain Molecules by Jonathon Simon Topological Chirality of Certain Molecules by Jonathon Simon Molecular Graphs as Topological Objects in Space by Jonathon Simon Molecular Graphs as Topological Objects in Space by Jonathon Simon Professor Steve Deckelman Professor Steve Deckelman