Fibered knots and potential counterexamples to the Property 2R and Slice-Ribbon Conjectures

Abstract

If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions.The simplest plausible counterexample to the Generalized Property R Conjecture could be a 2-component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to # 2 (S 1 × S 2 ). We exhibit a family of such links that are probably counterexamples to Generalized Property R. These links can be used to generate slice knots that are not known to be ribbon.

More Cappell–Shaneson spheres are standard

“…Thus, it is the mechanism underlying the endgames of [10; 4] and the examples of [12], as discussed in the Introduction. Alternatively, we can view @ˆas the S 1 -bundle over T 2 with Euler number˙1, and again see that the given diffeomorphism on this is isotopic to the identity.…”

“…(This appears in [12] in the context of Property R.) Akbulut's proof again introduces a 2-3 pair and runs the 2-handle around a torus to change m by 1. (In that case, one needs to strategically locate two circles that are unknotted in the surgered S 3 in order to find the whole torus -the author essentially did this in [10], but missed the crucial punch line!…”

On braided, banded surfaces and ribbon obstructions

“…3 These braid conjugacy class invariants are defined using the structure of the annular Khovanov-Lee complex of β S 3 as a (Z ⊕ Z)-filtered chain complex. 4 We recall here some of its relevant properties.…”

“…1]). 4 In [5], dt is defined as an invariant of oriented annular links, but it is noted there that the closure of a braid comes equipped with a standard orientation that we call the braid-like orientation, see [5,Sec. 3.2], and two braid-oriented braid closures are isotopic as annular links iff they are conjugate in B.…”

Characterizing Dehn surgeries on links via trisections

“…The possible counterexamples mentioned in the previous paragraph were produced by Gompf-Scharlemann-Thompson [GST10], building on work of Akbulut-Kirby [AK85]. We will call this family the GST links.…”