Khovanov homology is an unknot-detector

By P. Kronheimer, T. Mrowka. 2010

337 citations recorded by the source corpus.

We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement: an invariant that is already known to detect the unknot.

Subfields: Geometry and Topology, Neurology

View Paper: https://www.semanticscholar.org/paper/4195ae60dfd7e9f2392a54e30bf230bb32f001bd

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Citations in the corpus

Fivebranes and Knots (2011, background): “There have also been attempts [26, 27] to make the three- or fourdimensional symmetry of Khovanov homology manifest by extracting it from a special case or analog of Donaldson-Floer theory in four dimensions.”
Instantons and odd Khovanov homology (2014, methodology): “This approach was used by Kronheimer, Mrowka, Ozsváth and Szabó to prove a surgery exact sequence in the monopole case [22].”
Grid Homology for Knots and Links (2015, positive): “1 holds for Khovanov homology: Kronheimer and Mrowka [112] have shown that the unknot is the only knot whose Khovanov homology has dimension two over Q.”
The pillowcase and perturbations of traceless representations of knot groups (2013, methodology): “Summary of the results of Kronheimer-Mrowka. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 12.2.”
Tait colorings, and an instanton homology for webs and foams (2015, methodology): “This example is similar to the previous one, and was exploited in [15].”
The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality (2014, methodology): “Despite the simplicity of this description, the link invariant is quite powerful; for example, Rasmussen uses Khovanov homology to give an elegant proof of the Milnor conjecture [47], Kronheimer and Mrowka show that it detects the unknot [31], and 1”
Khovanov homology detects the trefoils (2018, background): “To prove that Khovanov homology detects the unknot, Kronheimer and Mrowka established in [KM11] a spectral sequence relating Khovanov homology and singular instanton knot homology, the latter of which assigns to a knot K ⊂ Y an abelian group I \(Y,K).”
Gauge theory and Rasmussen's invariant (2011, methodology): “The spectral sequence was defined in [8] using Z coefficients, and we need to adapt it now to the case of the local coefficient system Γ or Γ̂.”
Khovanov module and the detection of unlinks (2012, background): “It follows from [19] that each Li is an unknot, so L is an unlink.”
A link-splitting spectral sequence in Khovanov homology (2013, background): “The second, constructed by Kronheimer and Mrowka, begins at the Khovanov homology of a knot and converges to its instanton knot Floer homology [18].”
On the geography and botany of knot Floer homology (2014, background): “3), together with Kronheimer and Mrowka’s result that Khovanov homology detects the unknot [46].”