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Manifold Constructed from Two Tetrahedron: Figure Eight Knot Complement Jacob Clark [email protected] Knots ● Interestingly enough, this manifold is homeomorphic to the figure eight knot complement. ● ● ● 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 ● First, some definitions: ● ● Homeomorphism – is a continuous function with a continuous inverse. – preserves the properties of the topological space. Manifold – a topological space that on a small enough scale, it resembles the Euclidean space of a specific dimension, called the dimension of the manifold. – We shall concern ourselves with the notion of a 3-manifold, or a manifold that is locally homeomorphic to some shape in R3 ● To check this, we have to verify several conditions. Conditions for a Manifold ● ● 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 ● Points in the Faces ● Points on the Edges ● Vertices How many Vertices? ● There is one vertex Points on the faces Points on the Edges Neighborhood around Vertex Boundary of neighborhood around Vertex ● ● ● 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? ● Let's call our possible manifold M – v. ● Want to show M − v is homeomorphic to S3 − L, where L is the figure-eight knot. ● Lets prove this result: To begin, ● ● Begin with this cell complex K1 Recall a Cell complex is a topological space made up of cells ● ● Obtained from giving edges orientation and adding two bridges Clearly, this cell complex is made up of 0 cells and 1 cells – Namely, Vertices and edges ● To continue, we attach to this cell complex four 2 cells ● Lets call this K2 ● ● 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 ● So, there is such a homomorphism. ● This homeomorphism takes the 2 cells in K2 to the following 2 cells ● Thus, S3 – K2 is homeomorphic to the complement of the plane above. ● Which is two open 3 balls. – ● Claim proven! Now, looking at the open 3 balls as the interior of two 3 cells ● K2 extends to a cell complex K3 for S3 ● We still do not have a manifold yet. ● To get those nice 60 degree angles, we will observe the tetrahedron in the ball model. ● 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 ● Mark Lackenby ● Hyperbolic Manifolds – ● http://people.maths.ox.ac.uk/lackenby/hypox611.pdf William P. Thurston ● The Geometry and Topology of Three-Manifolds – http://library.msri.org/books/gt3m/