In the mathematical theory of knots

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...

, a Berge knot or doubly primitive knot is any member of a particular family of knots

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...

 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...

. A Berge knot K is defined by the conditions:

1. K lies on a genus two Heegaard surface S
2. in each handlebody
   Handlebody
   In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds...
   
    bound by S, K meets some meridian disc exactly once.

John Berge constructed these knots as a way of creating knots with 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...

 and classified all the Berge knots. Cameron Gordon

Cameron Gordon (mathematician)

Cameron Gordon is a Professor and Sid W. Richardson Foundation Regents Chair in the Department of mathematics at the University of Texas at Austin, known for his work in knot theory. Among his notable results is his work with Marc Culler, John Luecke, and Peter Shalen on the cyclic surgery theorem...

 conjectured these were the only knots admitting lens space surgeries. This is now known as the Berge conjecture

Berge conjecture

In the mathematical subject of knot theory, the Berge conjecture states that the only knots in the 3-sphere which admit lens space surgeries are Berge knots. The conjecture is named after John Berge....

.