We give an obstruction for genus one knots K, K' to have Gordian distance one by using the $0$th coefficient of the HOMFLY polynomials. As an application, we give a new constraint for a genus one knot to admit a (generalized) cosmetic crossing. Combining known results, we prove the (generalized) cosmetic crossing conjecture for genus one pretzel knots.

The author has been partially supported by JSPS KAKENHI Grant Number 19K03490 and 21H04428. He would like to thank H. Takioka for useful conversations, in particular for informing him of his computation of the zeroth coefficient polynomial of genus one pretzel knots.