On rational sliceness of Miyazaki's fibered, −amphicheiral knots

Abstract

We prove that fibered, −amphicheiral knots with irreducible Alexander polynomials are rationally slice. This contrasts with the result of Miyazaki that (2n, 1)-cables of these knots are not ribbon. We also show that the concordance invariants ν + and Υ from Heegaard Floer homology vanish for a class of knots that includes rationally slice knots. In particular, the ν +and Υ-invariants vanish for these cable knots.

Brieskorn spheres bounding rational balls

“…Recall that Σ(2, 3, 7) can be obtained by +1-surgery on the figure-eight knot, and in fact the construction in [FS84] can be modified to show that the figure-eight knot is rationally slice, that is, bounds a smooth disk in a rational homology ball bounded by S 3 (see [Cha07]). This fact was apparently also known by Kawauchi 1 [Kaw80], whose argument moreover generalizes to show that all strongly negative-amphicheiral knots are rationally slice (see [Kaw09], [KW16]). In Section 3 we give a handle proof that the figure-eight knot is rationally slice.…”

The $(2,1)$-cable of the figure-eight knot is not smoothly slice

“…Explicitly, in [Miy94, Example 2] Miyazaki showed that if K is a fibered, negative amphicheiral knot with irreducible Alexander polynomial, then the (2n, 1)-cable of K is not (homotopy) ribbon for any n = 0. On the other hand, these knots are known to be algebraically slice [Kaw80b,Proposition 4.1] and rationally slice [Kaw80a,Cha07,KW18]. 1 While such cables are generally believed not to be slice, the fact that no argument has appeared in the literature has left open the possibility that these generate counterexamples to the slice-ribbon conjecture.…”

Secondary Upsilon invariants of knots

“…This homomorphism provides bounds on the three-genus, the four-genus, and the concordance genus of knots. Since then, this invariant has been used to effectively address a range of problems; for a sampling, see [1,2,3,4,11,13,15,16].…”