In knot theory, the Milnor conjecture says that the slice genus of the ( p , q ) {\displaystyle (p,q)} torus knot is
( p − 1 ) ( q − 1 ) 2 . {\displaystyle {\frac {(p-1)(q-1)}{2}}.}
It is in a similar vein to the Thom conjecture. The conjecture (named after John Milnor) 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.
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