Topology : Betti Numbers
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Example Questions
Example Question #1 : Betti Numbers
What is the simplex in the zero dimension?
Vertex
Edge
Line
Object
Face
Vertex
In topology, Betti numbers represent the various dimensions of a topological space in regards to the number of holes that are present. There are four different simplices depending on the dimension: zero, one, two, three.
These are also referred to as:
Zero simplex refers to a point or vertex.
One simplex refers to a line or edge (which is connected by two points)
Two simplex refers to a face (which is connected by three lines)
Three simplex refers to the 3D object created by two faces.
Therefore the simplex in the zero dimension is known as a vertex.
Example Question #2 : Betti Numbers
What is the simplex in the second dimension?
Line
Face
Vertex
Edge
Space
Face
In topology, Betti numbers represent the various dimensions of a topological space in regards to the number of holes that are present. There are four different simplices depending on the dimension: zero, one, two, three.
These are also referred to as:
Zero simplex refers to a point or vertex.
One simplex refers to a line or edge (which is connected by two points)
Two simplex refers to a face (which is connected by three lines)
Three simplex refers to the 3D object created by two faces.
Therefore the simplex in the second dimension is known as a face. What is the simplex in the zero dimension?
Example Question #3 : Betti Numbers
What is the simplex in the first dimension?
Vertex
Edge
Face
Point
Object
Edge
In topology, Betti numbers represent the various dimensions of a topological space in regards to the number of holes that are present. There are four different simplices depending on the dimension: zero, one, two, three.
These are also referred to as:
Zero simplex refers to a point or vertex.
One simplex refers to a line or edge (which is connected by two points)
Two simplex refers to a face (which is connected by three lines)
Three simplex refers to the 3D object created by two faces.
Therefore the simplex in the first dimension is known as a edge.
Example Question #2 : Betti Numbers
Calculate the Euler Characteristic given the following Betti numbers.
The Euler Characteristic is calculated by the following equation.
This is also written as,
Recall that the Betti numbers represent the vertices, edges, and faces of the object.
In this particular question the Betti numbers are known therefore, substitute them into the formula to calculate the Euler Characteristic and solve.
Example Question #5 : Betti Numbers
Calculate the Euler Characteristic given the following Betti numbers.
The Euler Characteristic is calculated by the following equation.
This is also written as,
Recall that the Betti numbers represent the vertices, edges, and faces of the object.
In this particular question the Betti numbers are known therefore, substitute them into the formula to calculate the Euler Characteristic and solve.
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