I'd say Jazz might not exist if the saxaphone hasn't gotten invented- or if it did, it would be very different.
Actually, New Orleans purists frown on the saxophone, which was not part of the classic New Orleans "front line" which consisted of trumpet (or cornet), trombone, and clarinet. Yes, there was Sidney Bechet's soprano sax, but as Gunther Schuller wrote in *Early Jazz*, "Though technically speaking then a saxophone player, Bechet nevertheless belongs in a discussion of clarinetists because he played the soprano saxophone with a technique and style very much out of the New Orleans clarinet tradition. The reed player traditionally thinks of the soprano saxophone as a cousin to the clarinet..."
https://books.google.com/books?id=PfwfMTWBGgYC&pg=PA195 One traditionalist, Rex Harris, actually wrote that Coleman Hawkins would have been a better musician if he had stuck to the clarinet! Which, as someone remarked, "is a bit like saying that Mozart might have been OK if he had stuck to the harpsichord."
http://www.independent.co.uk/arts-e...worrying-and-love-the-curly-horn-1090211.html
Most jazz scholars today agree that this the purists are wrong, that saxophones (even apart from Bechet's soprano) did play a role in New Orleans jazz bands--consider the 1927 Sam Morgan recordings.
http://www.redhotjazz.com/sammorgen.html
In any event, it is true that is hard to conceive of later jazz styles--swing and especially bop and other "modern" styles--without the saxophone. The clarinet for some reason has never had the prominence in modern jazz that it had in New Orleans and swing.