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Khovanov homology is an unknot-detector. (English) Zbl 1241.57017

In this paper the authors prove that if the reduced Khovanov cohomology of a knot in the three-sphere is \(\mathbb{Z}\), then it is the unknot.
For a knot \(K\) let \(\text{Khr}(K)\) be the reduced Khovanov cohomology [M. Khovanov, “A categorification of the Jones polynomial”, Duke Math. J. 101, No. 3, 359-426 (2000; Zbl 0960.57005)], and \(\text{KHI}(K)\) be the instanton Floer homology associated with the balanced sutured manifold obtained from the knot complement by regarding two copies of meridian as the suture [P. Kronheimer and T. Mrowka, “Knots, sutures, and excision”, J. Differ. Geom. 84, No. 2, 301–364 (2010; Zbl 1208.57008)].
The authors introduce a link invariant \(\text{I}^{\natural}(K)\) by using connections with singularities. They show that with \(\mathbb{Q}\) coefficients, \(\text{I}^{\natural}(K)\) is isomorphic to \(\text{KHI}(K)\) by using Floer’s excision theorem [P. J. Braam and S. K. Donaldson, “Floer’s work on instanton homology, knots and surgery”, Hofer, Helmut (ed.) et al., The Floer memorial volume. Basel: Birkhäuser. Prog. Math. 133, 195–256 (1995; Zbl 0996.57516), A. Floer, “Instanton homology, surgery, and knots”, Geometry of low-dimensional manifolds. 1: Gauge theory and algebraic surfaces, Proc. Symp., Durham/UK 1989, Lond. Math. Soc. Lect. Note Ser. 150, 97–114 (1990; Zbl 0788.57008)]. Then they show that there exists a spectral sequence from \(\text{Khr}(\tilde{K})\) to \(\text{I}^{\natural}(K)\), where \(\tilde{K}\) is the mirror image of \(K\). Finally they use the fact that \(\text{KHI}\) is an unknot detector, see [Kronheimer and Mrowka, loc. cit.].

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Work of plenary speakers at ICM 2018

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R58 Floer homology
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References:

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