Why is the Braid Group Infinite?

In summary, the Braid Group is a mathematical concept that represents the movement and interactions of braids. It is considered infinite because it has an infinite number of elements or braids. This allows for a wide range of applications in mathematics, physics, and computer science. The Braid Group can be visualized through braid diagrams, and there are other mathematical groups that are infinite like the Braid Group, such as the symmetric group and the alternating group.
  • #1
gioialorusso
9
0
Hi,
can anyone explain me why (mathematically) the braid group is infinte? I guess it's infinite because you can do every braid you want and even if you braid two particles interchanging them twice in a clock (or counterclock) wise manner, (so you bring them back at the original positions), the system does not necessarily back to the same state. But has someone a good explanation of this fact?
Thank you,
gioia
 
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  • #2
B_1 is trivial, B_2 is infinite cyclic and B_{n-1} is a subgroup of B_n, so from the second on they are infinite.
 

1. What is the Braid Group?

The Braid Group is a mathematical concept that represents the movement and interactions of braids, which are arrangements of strands that can be twisted and crossed over each other.

2. Why is the Braid Group considered infinite?

The Braid Group is considered infinite because it has an infinite number of elements or braids. Each element in the Braid Group represents a unique arrangement of strands, and since there is no limit to the number of strands or the number of times they can be twisted and crossed over, the Braid Group is infinite.

3. How does the infinite nature of the Braid Group affect its applications?

The infinite nature of the Braid Group allows for a wide range of applications in mathematics, physics, and computer science. It can be used to study knot theory, topological quantum field theory, and cryptography, among other fields.

4. Can the Braid Group be visualized?

Yes, the Braid Group can be visualized through braid diagrams, which represent the movement and interactions of braids. These diagrams can help mathematicians study the properties and behavior of the Braid Group.

5. Are there any other mathematical groups that are infinite like the Braid Group?

Yes, there are other mathematical groups that are infinite, such as the symmetric group and the alternating group. These groups also have an infinite number of elements, but they have different properties and applications compared to the Braid Group.

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