And the verb you were looking for in the OP was “reciprocates”, not “equals”.
Division and multiplication are two different operations which may or may not be reciprocal. We call them reciprocal if “A*B = C” implies that “C/A = B” and that “C/B = A” and vice versa, for any combinations of numbers within the group (all three of A, B and C must always be in the group for all combos).
For real numbers (numbers positive or negative and with or without decimals, writtable in forms such as “2” or “456123.6123418534”), they are reciprocal except when one or both of A and B are 0.
For natural numbers (positive numbers with no decimals: 1, 2, 3…) they are not reciprocal: 2*3 = 6, 6/2 = 3, 6/2 = 3… but 3/2 = 1.5, which is not a natural number. We can find cases where the division of two members of the group is not in the group, therefore division can’t be reciprocal to multiplication (multiplication always produces results within the group).
There are other types of “numbers” in which the identity for addition wouldn’t be “0”, because they’re types of mathematical entities which aren’t writtable in the forms above. Vectors, matrices, geometrical figures… can all have multiplication and division defined, but they’re not even writtable the way all those entities in the previous examples are. Their own multiplication and division will produce other vectors, matrices, figures…
Bumping this because I came across this video whihc I think has a rather nice take on the question.
Also, I’m linking it because the guy doing the video and I had offices in the same corridor in a previous life. We were both PhD students at about the same time, and I also had a few conversations about geometry with him relating to my own work later on. Burkard was known for developing an algebra of juggling. Nice guy.
Francis Vaughan’s link cites a couple of articles which claim that every braid can be obtained as a trace of balls juggled in some pattern; I assume after you establish which patterns are juggleable that this is a consequence of some simple property of words in the braid group. The converse is trivially true, since if you trace the positions of the balls you obviously get some braid.
A propos of the original thread, in every (mathematical) group, like the group of braids, you can divide any element by any other element, because by definition every element has an inverse. It is only when you form the group algebra by allowing linear combinations of braids that zero shows up.