Milnor conjecture (knot theory)

For other uses, see Milnor conjecture (disambiguation).

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.^[1] Jacob Rasmussen later gave a purely combinatorial proof using Khovanov homology, by means of the s-invariant.^[2]