From zero surgeries to candidates for exotic definite four-manifolds

Abstract

One strategy for distinguishing smooth structures on closed 4-manifolds is to produce a knot K in S 3 that is slice in one smooth filling W of S 3 but not slice in some homeomorphic smooth filling W . In this paper we explore how 0-surgery homeomorphisms can be used to potentially construct exotic pairs of this form. In order to systematically generate a plethora of candidates for exotic pairs, we give a fully general construction of pairs of knots with the same zero surgeries. By computer experimentation, we find 5 topologically slice knots such that, if any of them were slice, we would obtain an exotic four-sphere. We also investigate the possibility of constructing exotic smooth structures on # n CP 2 in a similar fashion.

Categorical lifting of the Jones polynomial: a survey

“…As she notes, the s-invariant plays a special role here: other known smooth concordance invariants, like the Heegaard Floer analogue τ , would not work for this strategy. This proof, and the s-invariant, gives a possible attack on the smooth 4-dimensional Poincaré conjecture [119] (see also [49,118]). Functoriality of Khovanov homology means that it also gives an invariant of surfaces in R 4 .…”

Slicing knots in definite 4-manifolds

“…We now illustrate the use of the proposition above to find new examples of knots with finite u CP 2 and u CP 2 . Knots which are both positively slice and negatively slice were called biprojectively H-slice in [MP21].…”

“…Since then, numerous connections have been established between sliceness and fundamental open problems in 4-manifold topology, including the smooth Poincaré conjecture [FGMW10] and the exactness of the surgery sequence for topological 4-manifolds [CF84] (see also [KOPR21]). Recent work [MMP20, MMSW19,MP21] indicates that slicing knots not only in B 4 but in more general definite 4-manifolds may answer long-standing questions about the existence of exotic smooth structures in dimension four. As an example, [MP21, Theorem 1.4] provides a list of 23 knots, such that if any of them bounds a smooth, properly embedded, null-homologous disk in (# m CP 2 ) B4 , for some m, then there exists an exotic smooth structure on # m CP 2 .…”

Homology cobordism, knot concordance, and Heegaard Floer homology

“…If we could obstruct K from bounding a smoothly embedded disk in B 4 , then it follows that X cannot be diffeomorphic to S 4 . This approach was attempted in [FGMW10,MP21], but has yet to lead to a disproof of the conjecture.…”

“…1 The 3-dimensional homology cobordism group and the knot concordance group are fundamental structures in low-dimensional topology. The former played a key role in Manolescu's disproof of the high dimensional triangulation conjecture [Man16c], while the latter has the potential to shed light on the smooth 4-dimensional Poincaré conjecture (see, for example, [FGMW10,MP21]).…”