This is a very common question and generally the answer to this is no. The reason a topical will not show up on a drug test is that when you rub it into your skin it does not not penetrate deeply enough to enter the blood stream. This is also why topicals are known to treat localized vs generalized pain. However, if you use a transdermal patch, then it is more likely that you will test positive.
Some topicals do not use activated THC, and THC is what tests are usually looking for. We have several great articles on topicals. Here is one by Sweet Releaf:
I hope this helps! I am not a doctor and this represents my experience, research and opinion only.