
TL;DR
This paper extends the understanding of 2-torsion in instanton Floer homology to rational surgeries, establishing new obstructions for knots based on their instanton homology and small-surgery properties.
Contribution
It generalizes previous results from integral to rational surgeries and introduces new small-surgery obstructions related to $SU(2)$-abelian properties.
Findings
If $I^{lat}(S^3_r(K))$ is 2-torsion-free, then $K$ is an instanton L-space knot and $r>2g(K)-1.
Certain small surgeries imply $K$ is the unknot or trefoil, sharpening existing theorems.
Extension of 2-torsion analysis to all rational surgeries in instanton Floer homology.
Abstract
In our earlier work on -torsion in instanton Floer homology, we considered only integral surgeries on a knot and showed that the absence of -torsion forces to be fibered. The present paper extends the result to all rational surgeries. We prove that if the framed instanton homology is -torsion-free for some , then is an instanton L-space knot and . Leveraging this -torsion perspective, we also obtain new small-surgery obstructions: If either or is -abelian, then must be the unknot or the right-handed trefoil. This result sharpens the small--abelian surgery theorems of Kronheimer--Mrowka, Baldwin--Sivek, and Baldwin--Li--Sivek--Ye.
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