The issue is not that a circle has no beginning and end, but what led to the creation of the circle in the first place? It didn't materialize from nothingness, I drew it on a paper.
Rather than materialize from nothing, every event would be connected to the next and if you proceed along the events, you'll eventually see the same/similar events again. Sort of like a circularly linked list.
Except now you have a winding number and the two circles aren't equivalent everywhere.