Milnor conjecture (topology)
Encyclopedia
In knot theory
, the Milnor conjecture says that the slice genus
of the
torus knot
is
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.
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, the Milnor conjecture says that the slice genus
Slice genus
In mathematics, the slice genus of a smooth knot K in S3 is the least integer g such that K is the boundary of a connected, orientable 2-manifold S of genus g embedded in the 4-ball D4 bounded by S3.More precisely, if S is required to be smoothly embedded, then this integer g is the...
of the
torus knotTorus knot
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q...
is
It is in a similar vein to the Thom conjecture
Thom conjecture
In mathematics, a smooth algebraic curve C in the complex projective plane, of degree d, has genus given by the formulag = /2.The Thom conjecture, named after French mathematician René Thom, states that if \Sigma is any smoothly embedded connected curve representing the same class in homology as C,...
.
It was first proved by gauge theoretic
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
methods by Peter Kronheimer and Tomasz Mrowka
Tomasz Mrowka
Tomasz Mrowka is a Polish American mathematician. He has been the Singer Professor of Mathematics at Massachusetts Institute of Technology since 2010. A graduate of MIT, he received the Ph.D. from University of California, Berkeley in 1988 under the direction of Clifford Taubes and Robion Kirby...
. Jacob Rasmussen later gave a purely combinatorial proof
Combinatorial proof
In mathematics, the term combinatorial proof is often used to mean either of two types of proof of an identity in enumerative combinatorics that either states that two sets of combinatorial configurations, depending on one or more parameters, have the same number of elements , or gives a formula...
using Khovanov homology
Khovanov homology
In mathematics, Khovanov homology is an invariant of oriented knots and links that arises as the homology of a chain complex. It may be regarded as a categorification of the Jones polynomial....
, by means of the s-invariant.