From zero surgeries to candidates for exotic definite 4‐manifolds

Abstract

One strategy for distinguishing smooth structures on closed 4‐manifolds is to produce a knot in that is slice in one smooth filling of but not slice in some homeomorphic smooth filling . In this paper, we explore how 0‐surgery homeomorphisms can be used to potentially construct exotic pairs of this form. 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 five topologically slice knots such that, if any of them were slice, we would obtain an exotic 4‐sphere. We also investigate the possibility of constructing exotic smooth structures on in a similar fashion.

Trace embeddings from zero surgery homeomorphisms

“…The author would also like to thank his advisors Bob Gompf and John Luecke for their help and support. As noted in [17], some cases of the above results were already established by others. Dunfield and Gong showed that 𝐾 6 , … , 𝐾 21 are not slice using their program to compute twisted Alexander polynomial [4] and Kyle Hayden showed that 𝑍 1 is standard.…”

“…The classical trace embedding lemma asserts a knot 𝐾 is slice if and only if the zero trace 𝑋 0 (𝐾) smoothly embeds in 𝑆 4 . We have an analogous statement for a knot to be 𝐻-slice in 𝑋. Lemma 2.2 (H-slice trace embedding lemma, [17,Lemma 3.5]). A knot 𝐾 is 𝐻-slice in 𝑋 if and only if −𝑋 0 (𝐾) smoothly embeds in 𝑋 by an embedding that induces the zero map on second homology.…”

“…The exterior naturally has boundary 𝑆 3 0 (𝐾) and we can define 𝑋 ′ = 𝐸(𝐷) ∪ 𝜙 −𝑋 0 (𝐾 ′ ). 𝐾 ′ is 𝐻-slice in 𝑋 ′ by Lemma 2.2 and if 𝑋 is simply connected, 𝑋 ′ is also simply connected [17,Lemma 3.3] and therefore homeomorphic to 𝑋 by Freedman [7]. If 𝐾 ′ is not 𝐻-slice in 𝑋, then 𝐻-sliceness of 𝐾 ′ smoothly distinguishes 𝑋 ′ from 𝑋.…”

“…We will be working with framed slice disks in #𝑛ℂℙ 2 and in this setting, it is often more practical to construct the disks directly. To construct knots 𝐻-slice in some #𝑛ℂℙ 2 , one can use a full positive twist along algebraically zero strands as in [17,Lemma 3.2]. This generalizes to framed sliceness, but now we need to keep track of how the framing changes.…”

“…

Manolescu and Piccirillo (2023) recently initiated a program to construct an exotic 𝑆 4 or #𝑛ℂℙ 2 by using zero surgery homeomorphisms and Rasmussen's 𝑠-invariant.They find five knots that if any were slice, one could construct an exotic 𝑆 4 and disprove the Smooth 4dimensional Poincaré conjecture. We rule out this exciting possibility and show that these knots are not slice.

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