#### Transcript Topological Concepts and Machinery

Two Fold Coverings and We’re Not Talking About Plus Size Bikinis Presenters: Krista Joslin Carla Ranallo Cylde Tedrick Steve Egle Brian King Mark Herried Nate Zimmer Outline I) Topological Stereochemistry a) Molecular graphs as topological objects in space b) Topological Chirality and Achirality II) Molecular Moebius ladders a) Description and Background b) Statement of Simon’s 1986 Theorem III) Topological Concepts and Machinery a) Topological Spaces b) Manifolds c) Covering Spaces 1) 2-fold Coverings 2) 2-fold Branched Coverings 3) General Covering Spaces IV) Conclusion Topological Stereochemistry 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. Studies of synthesis, characterization and analysis of molecular structures that are topologically nontrivial. When can/cannot one embedded graph be “deformed” into another? What are the properties of embedded graphs that are preserved by deformation? 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. G is undirected if for two vertices, va and vb, the edge (va, vb) is equal to the edge (vb, va). Basically, all edges can flow in both directions. G is directed if for two vertices, va and vb, (va, vb) (vb, va). v1 v2 v4 v3 Graphical Representation (cont.) Two examples of graphical representations: * Undirected * Directed v1 v2 v1 v2 v4 v3 v4 v3 Undirected Graph Directed Graph Achirality Definition of Achirality A graph embedded in R3 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. Example of Achirality 2 – proponol is an achiral organic molecule. Key: Blue is carbon Yellow is CH3 Group Red is Oxygen White is Hydrogen Chirality Definition of Chirality A graph embedded in R3 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. Example of Chirality Our hands are chiral, they cannot be deformed into their mirror image. More examples of Chirality 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 One more example of Chirality 2 – Butanol is a chiral molecule. Key: Blue is carbon Yellow is Methyl Group Green is Ethyl Group Red is Oxygen White is Hydrogen Mathematical Models of Chirality 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. Homeomorphisms are either differentiable or piecewise linear. Ambient Isotopy Let A and B be contained in a set M , a mathematical model, which is a subset of Rn. 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. We can now say… Achiral molecules are ambient isotopic to their mirror image and Chiral molecules are ambient isotopic to their mirror image Topological Chirality/Achirality An embedded graph G R3 is topologically achiral if there exists an orientation reversing homeomorphism of (R3, G). If not, G is topologically chiral. 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. Two types of Homeomorphisms Let us consider h be a homeomorphism from R3 to itself… 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. Mobius Ladder A mobius ladder, Mn 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 ai are I and I + n. K is the loop of the mobius ladder Mn and a,…,an are the rungs of Mn. Mobius Ladder chirality Catenane (# 467) Left and right handed Mobius ladders Applications of Catenane Molecular Memory for Computers 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. 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. Works at room temperature. 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. Jon Simon’s Theorem (1986) 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. Jon Simon’s Theorem (1986) Used topological machinery to prove his theory. Such topological concepts and machinery used were topological spaces and covering spaces. Topological Concepts and Machinery Topological Spaces Topological Spaces A topology on a set X is a collection Τ if subsets of X have these properties: i) Ǿ, X € Τ ii) The union of the elements of any subcollection of Τ is in Τ iii) The intersection of the elements of any finite subcollection of Τ is in Τ Topological Spaces Continued A set X, for which a topology Τ has been specified is called a topological space written (X,T) If we have a topological space (X,T) and UcX, U€T, U is called an “open set” 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 Ǿ, X are both open, arbitrary unions of open sets are open and finite intersections of open sets are open Topological Spaces Continued Ex) X ~ a set, T = all subsets of X, this is called the discrete topology Ex) If {X,Ǿ} = T, this is called the trivial or indiscrete topology Ex) Let X= (a,b,c} then the discrete topology T ={ X, Ǿ, {a}, {b}, {c}, {a,b}, {a,c}, {b,c} } Topological Spaces Continued Basis Let (X, T) be a topological space. A basis B is a collection of subsets X, (called basis elements) such that: (1) xX B B, X B (2) if X B1 B2 (B1,B2 B) then B3 B X B3 B1 B2 Manifolds A Brief Introduction In essence, a manifold is a space that is locally like Rn, however lacking a preferred system of coordinates. Furthermore, a manifold can have global topological properties that distinguish it from the topologically trivial Rn. Definition Let M be a subset of Rp 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 Rp. 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 Rn or to the half-space R+n. 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 twomanifold, 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 (R3). 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 R3) whose center falls on x. By definition since V is open (in R3), the set U which is given by the intersection of sets T and V is also open. 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 R2. 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 R3. So what are some examples of manifolds? One-manifolds: line segments, lines, circles, and unions of these Two-manifolds (surfaces): Mobius strip, annulus, the surface of a sphere. Three-manifolds: a three dimesional sphere, three dimensional ball, torus (doughnut) Why the interest in manifolds? In general, manifolds have generated so much interest because they are easier to deal with than other subsets of Rn. Covering Spaces (2-fold coverings) Open sets and N-Manifolds Let M be a subset of Rp for some p. A subset U of M is said to be open in M if U=MV where V is an open set in Rp. 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 Rn or to the halfspace Rn+ = {(x1,…,xn) Rn | xn > 0}. Order of a homeomorphism Def: Let M be a subset of Rn and let h:MM be a homeomorphism. Let r be an natural number. Then hr 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. Covering Involution Let h:MM be an orientation preserving homeomorphism of order two. Projection Map Def: Let M and N be three manifolds. Let p:MN 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. Twofold Cover Let M and N be three manifolds, and let h:MM be a covering involution. Let p:MN 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. Twofold Branch Covering Twofold Branch Cover Explanation Consider the three-manifolds M and N. 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. 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. p(x) h(x) M N 2-D Example of a Twofold Branched Cover Let M denote a unit disk expressed in polar coordinates Define h: M -> M by h(r, q) = (r, q + 180), so that h rotates M by 180 Define p: M -> M by p(r, q) = (r,2q), 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 (r,q) h (r,q + 180) 2-D Example (cont.) Let M1 be the surface obtained by cutting M open along a single radius of the disk Let M2 be a copy of M1 Stretch M1 and M2 open, and disk M is obtained by gluing these two half-disks together Therefore h interchanges M1 and M2, and p sends each of M1 and M2 onto M M1 M2 M 2-D Example (cont.) 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 h(x) M1 M2 p(x) M General Covering Spaces 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 the P is called a covering map and E is said to be a covering space of B. 1 Projection (P) of E onto B E P U B Covering Involution H:E F jumping between “pancakes” 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, p/4 = 9p/4 = 13p/4, these are all mappings of the same point on the number unit circle. Conclusion Ambient isotopy and homeomorphisms are graphical representations of chirality/achirality Achirality/chirality are important concepts to the fields of chemistry and biology With machinery gained from Jon Simon’s proof such as: Topological spaces, manifolds, and coverings spaces we gain new knowledge on how to solve present problems today, especially dealing with nanotechnology. Sources www.cosmiverse.com “When Chemistry Meets Topology” by Erica Flapan “Topological Chirality of Certain Molecules” by Jonathon Simon “Molecular Graphs as Topological Objects in Space” by Jonathon Simon Professor Steve Deckelman