Short answer yes, everything can be produced but artificial topaz does not seem to be being produced.
I took a look at only the precious stones because if there is no value in producing a synthetic, then why produce it? According to infoplease, "The precious stones are diamond; some forms of corundum (ruby, sapphire, Oriental emerald, Oriental topaz, and Oriental amethyst); and emerald." Based on that, "Among the synthetic gems produced commercially are rubies, sapphires, emeralds, and spinels. Diamonds of gem quality have also been manufactured."
Read more: gem: Types of Gemstones | Infoplease.com
http://www.infoplease.com/encyclopedia/science/gem-ornamental-mineral-or-organic-substance-types-gemstones.html#ixzz2TlIkhUWm
According to several websites, synthetic amethyst is fairly common.
According to Gemology.com forum, "As far as I know Synthetic Topaz doesn't exist and probably never will. There is a ton of Irradiated blue and coated (vapor deposition?) topaz on the market for cheap because Topaz is very plentiful, even in large sizes. I mean, I can go to Utah and literally collect pounds of facet-grade topaz in one weekend- for free! Some of the more desirable colors like the oranges and sherry-colored stones might be harder to come by (esp. large sizes) but colorless and extremely pale colors aren't too hard to find, and those are the ones treated. If I am wrong someone please correct " Later in the forum, someone posted, "The Institute of Experimental Mineralogy of Chernologovka, Russia published a report in 2000 where they succeded in growing gemquality topaz in a hydrothermal process. It was done on experimental basis and I do not know if there is any production going on anylonger"
