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)  xX  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=M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 halfspace Rn+ = {(x1,…,xn)  Rn | xn > 0}.
Order of a homeomorphism
Def: Let M be a subset of Rn and let
h:MM 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:MM be an orientation preserving
homeomorphism of order two.
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.
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.
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