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.
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