In the early XXth century, Alexander showed that every knot is the closure of a braid, and Markov gave necessary and sufficient conditions for two braids to have the same closure.
My question is: why don't we represent knots by its braid word then? Why was Conway notation so celebrated? (It is often said that it is the first notation for knots that was i) systematic and ii) lets one deduce properties of the knot at first glance. But this is also true for the braid words, isn't it?)
I think I'm maybe not understanding what the hidden problem behind using braids as a notation system is. For instance, there's a 2004 paper by Gittings on using braids as a knotational system — why did we have to wait since Markov's Theorem to 2004 to get this knotational system?
I only recently started reading into knot theory, so I might be being naive.