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The Igros Moshe concerning cotton tzitzis addresses the question of synthetics as well; if I recall, he points to the Rambam who says there's no mitzva of tzitzis on leather garments. Usually a leather garment isn't woven; now you could cut strands of leather and weave them, but I think R' Moshe implies that because leather starts off as a sheet and not fibers, there's no mitzva even if you weave it. I think he suggests the same for synthetics, as they can be made in one giant sheet and then cut and woven (as opposed to cotton, which starts off as fibers). But that should be a starting point.