An obstruction of Gordian distance one and cosmetic crossings for genus one knots

Abstract

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