Why is the Braid Group Infinite?

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SUMMARY

The braid group is infinite due to its structure, where B_1 is trivial, B_2 is infinite cyclic, and B_{n-1} is a subgroup of B_n for n ≥ 2. This means that as the number of strands increases, the complexity of the braids also increases, leading to an infinite number of distinct configurations. The mathematical reasoning behind this is rooted in the non-commutative nature of braiding, where even returning particles to their original positions does not guarantee a return to the same state.

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gioialorusso
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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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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.
 

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