Download Manifold Constructed from Two Tetrahedron: Figure Eight

Manifold Constructed from Two
Tetrahedron: Figure Eight Knot
Complement
Jacob Clark
[email protected]
Knots
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Interestingly enough, this manifold is homeomorphic to the figure eight knot
complement.
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A mathematical knot is somewhat different from what one would usually
consider a knot, i.e of as a piece of string with free ends.
The knots in mathematics are always considered to be closed loops.
Formally, a knot is an embedding of a circle in 3-dimensional Euclidean
space, R3, considered up to continuous deformations.
Definitions
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First, some definitions:
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Homeomorphism
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is a continuous function with a continuous inverse.
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preserves the properties of the topological space.
Manifold
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a topological space that on a small enough scale, it resembles the
Euclidean space of a specific dimension, called the dimension of the
manifold.
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We shall concern ourselves with the notion of a 3-manifold, or a
manifold that is locally homeomorphic to some shape in R3
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To check this, we have to verify several conditions.
Conditions for a Manifold
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Now, to show this is a manifold, we must check the 3 conditions, that an
open ball surrounds every point in the structure.
So, we must check
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Points in the Faces
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Points on the Edges
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Vertices
How many Vertices?
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There is one vertex
Points on the faces
Points on the Edges
Neighborhood around Vertex
Boundary of neighborhood around
Vertex
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For something to be a manifold, the vertex must have a neighborhood that is
homeomorphic to the open ball in R3, so bounded by a sphere!
Thus, the Complex is not a manifold, if the vertex is included.
But, if we remove the vertex, if we can find a place where our dihedral
angles are 60 degrees, we have a manifold.
What of the Figure Eight Knot?
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Let's call our possible manifold M – v.
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Want to show M − v is homeomorphic to S3 − L, where L is the figure-eight knot.
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Lets prove this result:
To begin,
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Begin with this cell complex K1
Recall a Cell complex is a topological
space made up of cells
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Obtained from giving edges
orientation and adding two
bridges
Clearly, this cell complex is made
up of 0 cells and 1 cells
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Namely, Vertices and edges
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To continue, we attach to this cell complex four 2 cells
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Lets call this K2
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I now claim that S3 – K2 is homeomorphic to two 3-balls.
To prove this easily, let us first show that there is a homeomorphism
between S3 – K1 and the complement of the cell below, denoted K11
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So, there is such a homomorphism.
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This homeomorphism takes the 2 cells in K2 to the following 2 cells
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Thus, S3 – K2 is homeomorphic to the complement of the plane above.
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Which is two open 3 balls.
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Claim proven!
Now, looking at the open 3 balls as the interior of two 3 cells
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K2 extends to a cell complex K3 for S3
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We still do not have a manifold yet.
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To get those nice 60 degree angles, we will observe the tetrahedron in the
ball model.
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The Ball model for Hyperbolic space is the inside of a ball, with one point
at infinity.
Visualization of Figure Eight Knot
Complement
http://www.lbl.gov/Science-Articles/Archive/assets/images/2003/Nov-03-2003/figure_eight_knot.jpg
Sources
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Mark Lackenby
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Hyperbolic Manifolds
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http://people.maths.ox.ac.uk/lackenby/hypox611.pdf
William P. Thurston
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The Geometry and Topology of Three-Manifolds
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http://library.msri.org/books/gt3m/