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This paper begins with a review of the extension of A'Campo's divides to the more general construction of oriented divides, using the same disk model of S3. It is shown that all links in S3 can be represented by oriented divides (in contrast with divide links) and simple methods are presented for moving between links and their oriented divide presentations. Using the properties of the disk model, oriented divides are extended by three extra equivalence rules, and it is shown that this extended theory of oriented divides is equivalent to knot theory in S3. Fueled by this result, the concept of oriented divide braid is introduced, together with a group construction which parallels the classical braid group representation of geometric braids. This leads to results analagous to Alexander's well-known theorem concerning braid presentations, and also the Markov theorem.

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