A crossing change in a diagram of a knot K is called cosmetic if the resulting diagram also represents K. The cosmetic crossing conjecture [10, Problem 1.58] posits that for any knot K, the only cosmetic crossing changes are nugatory, i.e. there exists an embedded 2-sphere in S3 which intersects K only at the two points of the relevant crossing. Conversely, it is not hard to see that any nugatory crossing change is cosmetic.
The cosmetic crossing conjecture has been proven for:
• Two-bridge knots, by Torisu [14].
• Fibered knots, by Kalfagianni [9].
• Genus one knots with non-trivial Alexander polynomial, by Ito following work of Balm, Friedl, Kalfagianni and Powell [1, 7].
• Whitehead doubles of prime, non-cable knots, by Balm and Kalfagianni [3].
• Special alternating knots, by Boninger using work of Lidman and Moore [5,12].
There are also significant obstructions to cosmetic crossing changes for alternating knots (and more generally, for knots with thin Khovanov homology). Lidman and Moore [12] gave a condition on the homology of the branched double-cover of the knot, and Boninger [4] gave a condition on the Alexander polynomial—both obstructions imply that alternating knots with square-free determinant do not admit cosmetic crossing changes. A third result was proven by Ito [6].
For more results regarding the cosmetic crossing conjecture, see [2,8,11,13,15]. A knot K in the table is given a U if the cosmetic crossing conjecture is open for K, or in other words if it is unknown whether K admits a non-nugatory cosmetic crossing change.
(Thanks go to Joe Boninger for compiling the initial KnotInfo data for cosmetic crossing changes and for initially writing this descriptive page.)
10128
Ref. [12]
1067
Ref. [6]
[1] Cheryl Balm, Stefan Friedl, Efstratia Kalfagianni, and Mark Powell, Cosmetic crossings and Seifert matrices, Comm. Anal. Geom. 20 (2012), no. 2, 235-253.
[2] Cheryl Jaeger Balm, Generalized crossing changes in satellite knots, Proc. Amer. Math. Soc. 143 (2015), no. 1, 447-458.
[3] Cheryl Jaeger Balm and Efstratia Kalfagianni, Knots without cosmetic crossings, Topology Appl. 207 (2016), 33-42.
[4] Joe Boninger, An Alexander Polynomial Obstruction to Cosmetic Crossing Changes (preprint).
[5] Joe Boninger, On the cosmetic crossing conjecture for special alternating links, Proc. Amer. Math. Soc. Ser. B 10 (2023), 288-295.
[6] Tetsuya Ito, Applications of the Casson-Walker invariant to the knot complement and the cosmetic crossing conjectures, Geom. Dedicata 216 (2022), no. 6, Paper No. 63, 15.
[7] Tetsuya Ito, Cosmetic crossing conjecture for genus one knots with non-trivial Alexander polynomial, Proc. Amer. Math. Soc. 150 (2022), no. 2, 871-876.
[8] Tetsuya Ito, An obstruction of Gordian distance one and cosmetic crossings for genus one knots, New York J. Math. 28 (2022), 175-181.
[9] Efstratia Kalfagianni, Cosmetic crossing changes of fibered knots, J. Reine Angew. Math. 669 (2012), 151-164.
[10] Rob Kirby, Problems in low dimensional manifold theory, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 273-312.
[11] Artem Kotelskiy, Tye Lidman, Allison H. Moore, Liam Watson, Claudius Zibrowius, Cosmetic operations and Khovanov multicurves. Math. Ann. 389, 2903-2930 (2024).
[12] Tye Lidman and Allison H. Moore, Cosmetic surgery in L-spaces and nugatory crossings, Trans. Amer. Math. Soc. 369 (2017), no. 5, 3639-3654.
[13] Allison H. Moore, Symmetric unions without cosmetic crossing changes, Advances in the mathematical sciences, Assoc. Women Math. Ser., vol. 6, Springer, [Cham], 2016, pp. 103-116.
[14] Ichiro Torisu, On nugatory crossings for knots, Topology Appl. 92 (1999), no. 2, 119-129
[15] Joshua Wang, The cosmetic crossing conjecture for split links, Geom. Topol. 26 (2022), no. 7, 2941-3053.