Solution: The set
is finite when and infinite
when . Let’s see why this is
true. First consider the case when . There is only one -braid, and it is
This means that there is only one element in , and so is finite. Next, consider the case
when . To see why is infinite, consider the following
-braids:
In all of these -braids, we see
that the second string wraps around the first string a various number of
times. From left to right and top to bottom, we see that the number of
wraps increases. From left to right, the -braids in the top row have 0 wraps, 1
wrap, and 2 wraps. From left to right, the -braid in the bottom row have 3 wraps, 4
wraps, and 5 wraps. We can keep increasing the number of these wraps
forever, and so there are an infinite number of -braids.
Lastly, consider the case when . We will use to show
that there are an infinite number of -braids. We construct -braids as follows. Take the last strings and dots to be

Then, we can think of the the remaining string connections (aka top 2
strings and pairs of dots) as a -braid. In other words, we have an
"embedding" of into . By varying the top two string
connections through the elements of , we generate an infinite number of
-braids (since is infinite). So, is infinite. Here are some concrete
examples of -braids constructed in
this way:
We see that the top two string connections are just the -braids from above.