Consider the the braid group of $n$ strands on the 2-sphere. Visually, this is a braid between two concentric circles. How many different braids are there for a given $n$?
I have tried drawing the Cayley diagram for $n=3$, which turns out to have order 12. Here is the diagram, with generators $a,b$ having inverses $A,B$, with identity $e$. This shows 2 different layouts of the same graph.
I have also tried drawing the Cayley diagram for $n=4$, but it seems to be much larger.
Reminder of the definitions: The braid group on the 2-sphere with $n$ strands is generated by $s_1,\dots,s_{n-1}$ each representing a single-crossing braid between adjacent strands; this obeys the following relations:
- $s_is_{i+1}s_i = s_{i+1}s_is_{i+1}$ (the "braid identity")
- $s_is_j = s_js_i$ unless $i =j-1$ or $i=j+1$ (non-adjacent crossings commute)
- $s_1s_2\dots s_{n-1}~s_{n-1}\dots s_2s_1 = e$ (a special identity for the 2-sphere)
Best Answer
There is a good reason for why $n=4$ seems to be much larger: the braid group with $4$ strands on the $2$-sphere is infinite!
In fact, it is known that for $n \geq 4$, the braid group with $n$ strands on the $2$-sphere is infinite. For a reference, see "The braid groups of $E^{2}$ and $S^{2}$", by Fadwell and Buskirk.