In the mathematical subject of knot theory

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 Berge conjecture states that the only knot

Knot (mathematics)

In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...

s in the 3-sphere

3-sphere

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...

 which admit lens space

Lens space

A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions....

 surgeries

Dehn surgery

In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...

 are Berge knot

Berge knot

In the mathematical theory of knots, a Berge knot or doubly primitive knot is any member of a particular family of knots in the 3-sphere. A Berge knot K is defined by the conditions:# K lies on a genus two Heegaard surface S...

s. The conjecture (and family of Berge knots) is named after John Berge.

Progress on the conjecture has been slow. Recently Yi Ni proved that if a knot admits a lens space surgery, then it is fibered

Fibered knot

A knot or link Kin the 3-dimensional sphere S^3 is called fibered or fibred if there is a 1-parameter family F_t of Seifert surfaces for K, where the parameter t runs through the points of the unit circle S^1, such that if s is not equal to t...

. Berge knots are fibered.

External links

Two blog

Blog

A blog is a type of website or part of a website supposed to be updated with new content from time to time. Blogs are usually maintained by an individual with regular entries of commentary, descriptions of events, or other material such as graphics or video. Entries are commonly displayed in...

posts in the weblog "Low Dimensional Topology - Recent Progress and Open Problems"
related to the Berge conjecture:

The Berge conjecture, by Jesse Johnson

Knot complements covering knot complements by Ken Baker