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

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

2. Presenters:Krista JoslinCarla RanalloCylde TedrickSteve EgleBrian KingMark HerriedNate Zimmer

3. 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

4. Topological Stereochemistry

5. 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?

6. 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

7. Graphical Representation (cont.) • Two examples of graphical representations: * Undirected * Directed v1 v2 v1 v2 v4 v3 v4 v3 Undirected Graph Directed Graph

8. Achirality

9. 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.

10. Example of Achirality • 2 – proponol is an achiral organic molecule. Key: Blue is carbon Yellow is CH3 Group Red is Oxygen White is Hydrogen

11. Chirality

12. 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.

13. Example of Chirality • Our hands are chiral, they cannot be deformed into their mirror image.

14. 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

15. 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

16. Mathematical Models of Chirality

17. 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.

18. 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.

19. We can now say… • Achiral molecules are ambient isotopic to their mirror image and • Chiral molecules are ambient isotopic to their mirror image

20. 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.

21. 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.

22. 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.

23. 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.

24. Mobius Ladder chirality

25. Catenane (# 467) • Left and right handed Mobius ladders

26. Applications of Catenane • Molecular Memory for Computers

27. 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.

28. 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.

29. MolecularMemory • 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.

30. 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.

31. Jon Simon’s Theorem (1986) • Used topological machinery to prove his theory. Such topological concepts and machinery used were topological spaces and covering spaces.

32. Topological Conceptsand Machinery

33. Topological Spaces

34. Topological Spaces • A topology on a set X is a collection Τif subsets of X have these properties: • i) Ǿ, X € Τ • ii) The union of the elements of any subcollection of Τis in Τ • iii) The intersection of the elements of any finite subcollection of Τ is in Τ

35. 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”

36. 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

37. 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} }

38. 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)  xX  B  B, X  B • (2) if X  B1  B2 (B1,B2  B) then B3  B X B3  B1 B2

39. Manifolds

40. 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.

41. 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.

42. 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 (R3).

43. 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.

44. 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.

45. 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.

46. So what are some examples of manifolds? • One-manifolds: line segments, lines, circles, and unions of these

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

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

49. 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.

50. Covering Spaces (2-fold coverings)