There was definitely a connection between the fertile crescent, Indus Valley and East Asia but I doubt that farming evolved independently in multiple areas of the world.
More likely farmers from fertile crescent migrated to Indus Valley and from there to China.
Agriculture did evolve independently in various parts of the world. The American continent and New Guinea are proof enough. As for China, Cui et al. 2013 tested 47 Neolithic Y-DNA samples from the West Liao River Valley, Northeast China and the haplogroups turned out to be exclusively East Asian (N1, C/C3e and O3a). In any case there is no evidence of migration from West or South Asia to China in Early Neolithic times, either from ancient DNA or from archaeology. Chinese pottery also pre-dates West Asian one by over 10,000 years.