On this subject I love this strangeloop talk [1]: the right notation allowed John Conway to solve in an afternoon a knot problem that took another mathematician (Little) 6 years to solve!
Another example is juggling notation, that allowed not only the sharing of patterns but the discovery of new ones [2].
1: https://www.youtube.com/watch?v=Wahc9Ocka1g
2: https://en.wikipedia.org/wiki/Juggling_notation