syntactic analysis - don't and don't in the same sentence? What was the authors point?

Reading an article, I came across this sentence which I can't seem to make sense of. The author uses a double negative but I'm not sure what the point is. Could this just be poor grammar?

Here is the sentence: "I don’t have some gene that others don’t"

Sentence for full context: I don’t have some gene that others don’t; I definitely haven’t been lucky.

If anyone would answer and help me understand, I would really appreciate it!