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Heegaard Floer homology and knots determined by their complements. (English) Zbl 1386.57009

The cosmetic surgery conjecture claims that there is only one slope on a knot that yields the same manifold. More precisely, let \(K\) be a knot in a closed connected orientable \(3\)-manifold whose exterior is irreducible and not homeomorphic to a solid torus. If two slopes \(r\) and \(s\) yield the same manifold under an orientation-preserving homeomorphism, then \(r\) and \(s\) are equivalent, that is, there is a homeomorphism of the knot exterior sending \(r\) to \(s\).
Although the cosmetic surgery conjecture is open in general, Y. Ni and Z. Wu [J. Reine Angew. Math. 706, 1–17 (2015; Zbl 1328.57010)] made significant progress for knots in the \(3\)-sphere by using Heegaard Floer homology. In particular, they showed that at most two slopes on a knot can yield the same oriented manifold and they are negatives of each other.
The first result of the paper under review gives an upper bound for the number of slopes on a knot in a homology \(3\)-sphere that yield the same manifold. Here is the statement. Let \(K\) be a knot in a homology \(3\)-sphere \(Y\). If two slopes \(p/q_1\) and \(p/q_2\) yield the same rational homology \(3\)-sphere \(Z\) such that \(|H_1(Z)|=p\) and \(p\) does not divide the Euler characteristic \(\chi(HF_{\mathrm{red}}(Z))\) of the reduced Heegaard Floer homology of \(Z\), then there is no multiple of \(p\) between \(q_1\) and \(q_2\). Hence there are at most \(\phi(p)\) slopes on \(K\) yielding \(Z\), where \(\phi\) is the Euler totient function. Also, infinitely many rational homology spheres which satisfy the condition of the above theorem are given.
As the second result, the author shows that if \(p/q\) surgery on a knot in a non-\(L\)-space homology sphere \(Y\) yields a rational homology sphere \(Z\) and \(|q|\) is bigger than a certain integer determined by the reduced Heegaard Floer homology of \(Y\) and \(Z\), then \(K\) satisfies several properties. The case where \(Y\) is an \(L\)-space homology sphere is covered in another paper by the author [Algebr. Geom. Topol. 17, No. 4, 1917–1951 (2017; Zbl 1380.57014)].
The second half of the paper discusses the knot complement conjecture. For example, it is shown that knots of genus larger than one in the Brieskorn sphere of type \((2,3,7)\) are determined by their complements. Also, the surgery characterization of the unknot for null-homologous knots in \(L\)-spaces is proved. This is a generalization of the case where the ambient manifold is the \(3\)-sphere, see P. Kronheimer et al. [Ann. Math. (2) 165, No. 2, 457–546 (2007; Zbl 1204.57038)]. Hence null-homologous knots in \(L\)-spaces are determined by their complements. Finally, if \(p\) is square-free, then all knots whose exteriors are not solid tori in a lens space \(L(p,q)\) are shown to be determined by their complements.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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References:

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