knot theory In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...

, the Milnor conjecture says that the

slice genus In mathematics, the slice genus of a smooth knot ''K'' in ''S''3 (sometimes called its Murasugi genus or 4-ball genus) is the least integer g such that ''K'' is the boundary of a connected, orientable 2-manifold ''S'' of genus ''g'' properly embed ...

of the

(p, q)

torus knot In knot theory, a torus knot is a special kind of knot (mathematics), knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link (knot theory), link which lies on the surface of a torus in the same way. Each t ...

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 Peter Benedict Kronheimer (born 1963) is a British mathematician, known for his work on gauge theory and its applications to 3- and 4-dimensional topology. He is William Caspar Graustein Professor of Mathematics at Harvard University and former ...

and

Tomasz Mrowka Tomasz Mrowka (born September 8, 1961) is an American mathematician specializing in differential geometry and gauge theory. He is the Singer Professor of Mathematics and former head of the Department of Mathematics at the Massachusetts Institut ...

. Jacob Rasmussen later gave a purely

combinatorial proof In mathematics, the term ''combinatorial proof'' is often used to mean either of two types of mathematical proof: * A proof by double counting. A combinatorial identity is proven by counting the number of elements of some carefully chosen set ...

using

Khovanov homology In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial. It was developed in the late 1990s by Mikhail Khovanov, then at ...

, by means of the s-invariant..

References

Geometric topology Knot theory 4-manifolds Conjectures that have been proved {{knottheory-stub