Milnor conjecture (topology)

For Milnor's conjecture about K-theory, see Milnor conjecture.

In knot theory, the Milnor conjecture says that the slice genus of the (p,q) torus knot is

(p − 1)(q − 1) / 2.

It is in a similar vein to the Thom conjecture.

It was first proved by gauge theoretic methods by Peter Kronheimer and Tomasz Mrowka.^[1] Jacob Rasmussen later gave a purely combinatorial proof using Khovanov homology, by means of the s-invariant.^[2]

References

1. ^ Kronheimer, P. B.; Mrowka, T. S. (1993), "Gauge theory for embedded surfaces, I", Topology 32 (4): 773–826, doi:10.1016/0040-9383(93)90051-V .
2. ^ Rasmussen, Jacob A. (2004). "Khovanov homology and the slice genus". arXiv:math.GT/0402131. .

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