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Khovanov homology and the slice genus. (English) Zbl 1211.57009

In this groundbreaking and important paper, Rasmussen uses Khovanov homology to define a smooth knot concordance invariant \(s\) and proves that \(s\) gives a lower bound for the smooth slice genus. He shows that \(s\) is equal to the knot signature for alternating knots, and more interestingly that \(s\) is equal to the genus and the slice genus for knots admitting a diagram containing only positive crossings. This gives a combinatorial proof of Milnor’s conjecture that the slice genus (and unknotting number) of the \(T_{p,q}\) torus knot is \((p-1)(q-1)/2\). This was first proved by P. B. Kronheimer and T. S. Mrowka [Topology 32, No. 4, 773–826 (1993; Zbl 0799.57007)] using instanton gauge theory.
The definition of \(s\) uses E. S. Lee’s proof [Adv. Math. 197, No. 2, 554–586 (2005; Zbl 1080.57015)] that Khovanov homology is the \(E^1\) term of a spectral sequence converging to \(\mathbb Q\oplus\mathbb Q\) and is analagous to the definition of the knot concordance invariant \(\tau\) of Ozsváth and Szabó. The author conjectures in the paper under review that in fact \(s=2\tau\); counterexamples have since been found by M. Hedden and P. Ording [Am. J. Math. 130, No. 2, 441–453 (2008; Zbl 1139.57012)].

MSC:

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

Software:

Khoho; Knot Atlas
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Full Text: DOI arXiv

References:

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