The Conway knot is not slice. (English) Zbl 1471.57011

This paper is yet another example of the endless variety of knots and the difficulty that mathematicians may have in distinguishing them. In [in: Comput. Probl. abstract Algebra, Proc. Conf. Oxford 1967, 329–358 (1970; Zbl 0202.54703)], J. H. Conway published a list of 548 prime knots with eleven crossings, missing just four non-alternating examples (identified by Alain Caudron in 1979, cf. [A. Caudron, Publ. Math. Orsay 82–04, 336 p. (1982; Zbl 0505.57002)]) but including a pair with trivial Alexander polynomial (and identical Jones, Homflypt and Kaufmann polynomials). One (a.k.a. 11n34) is a mutant of the other (a.k.a. 11n42), previously discovered by Kinoshita and Terasaka. In [Math. Comput. 25, 603–619 (1971; Zbl 0224.55003)] R. Riley (who first discovered hyperbolic knots) successfully distinguished these two knots by reference to the homology of their branched non-cyclic covering spaces. In [Mem. Am. Math. Soc. 339, 1–98 (1986; Zbl 0585.57003)] D. Gabai did the same by reference to their Seifert genera. This author now offers a third method for distinguishing this pair of knots, showing also that they give the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot, and completing the list of slice knots under 13 crossings. (Her proof is also described by J. Hom [Bull. Am. Math. Soc., New Ser. 59, No. 1, 19–29 (2022; Zbl 1477.57004)]). The fact that it took so long for this non-sliceness to be determined is a tribute to the ability of certain knots to withstand targeted scrutiny for quite some time. For example, it took more than 73 years for someone to see that the Perko pair was just one knot. See also the comment on Stoimenow’s “Knot data tables” website that “particular care is needed with (Rolfsen’s) \(10_{71}\), for it is not amphicheiral, yet no invariant tabulated here (and computable with KnotScape) can distinguish it from its mirror image.”


57K10 Knot theory
57R65 Surgery and handlebodies
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