New York Journal of Mathematics
Volume 28 (2022), 175-181


Tetsuya Ito

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

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Published: January 26, 2022.
Keywords: Gordian distance, cosmetic crossing conjecture, HOMFLY polynomial.
Subject: 57K10, 57K31.

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.

Author information

Tetsuya Ito:
Department of Mathematics
Kyoto University
Kyoto 606-8502, JAPAN