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In knot theory, the Milnor conjecture says that the slice genus of the ${\displaystyle (p,q)}$ torus knot is

${\displaystyle (p-1)(q-1)/2.}$

It is in a similar vein to the Thom conjecture.

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

References

1. Kronheimer, P. B.; Mrowka, T. S. (1993), "Gauge theory for embedded surfaces, I" (PDF), 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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