In the
mathematical theory of knots, a Berge knot (named after mathematician John Berge) or doubly primitive knot is any member of a particular family of
knots
A knot is a fastening in rope or interwoven lines.
Knot may also refer to:
Places
* Knot, Nancowry, a village in India
Archaeology
* Knot of Isis (tyet), symbol of welfare/life.
* Minoan snake goddess figurines#Sacral knot
Arts, entertainme ...
in the
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
. A Berge knot ''K'' is defined by the conditions:
# ''K'' lies on a
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
two
Heegaard surface ''S''
# in each
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. Handles ...
bound by ''S'', ''K'' meets some meridian disc exactly once.
John Berge constructed these knots as a way of creating knots with
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.
In the 3-manifold case, a lens space can be visualize ...
surgeries
Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical specialty that uses operative manual and instrumental techniques on a person to investigate or treat a pat ...
and classified all the Berge knots.
Cameron Gordon conjectured these were the only knots admitting lens space surgeries. This is now known as the
Berge conjecture.
Berge conjecture
The Berge conjecture states that the only
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
s in the
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...
which admit
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.
In the 3-manifold case, a lens space can be visualize ...
surgeries
Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical specialty that uses operative manual and instrumental techniques on a person to investigate or treat a pat ...
are Berge knots. 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. Subsequently,
Joshua Greene showed that the lens spaces which are realized by surgery on a knot in the 3-sphere are precisely the lens spaces arising from surgery along the Berge knots.
Further reading
Knots
*.
*.
*.
Conjecture
*.
*.
*.
External links
Two
blog
A blog (a truncation of "weblog") is a discussion or informational website published on the World Wide Web consisting of discrete, often informal diary-style text entries (posts). Posts are typically displayed in reverse chronological order ...
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 complementsby Ken Baker
{{Knot theory, state=collapsed
Knots and links
3-manifolds