All Topics  
Unknot

 

 
   Email Print
   Bookmark   Link
 

 

Unknot
 
 
  Topics Unknot Topic Home

The unknot arises in the mathematical theory of knots
Knot theory

In mathematics, knot theory is the area of topology that studies knot s. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs drastically in that the ends are joined together to prevent it from becoming undone....
. Intuitively, the unknot is a closed loop of rope without a knot
Knot

A knot is a method for fastening or securing linear material such as rope by tying or interweaving. It may consist of a length of one or more segments of rope, string, webbing, twine, strap, or even chain interwoven such that the line can bind to itself or to some other object?the "load"....
 in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the circle as a geometrically round circle. The unknot is also called the trivial knot. An unknot is the identity element
Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
 with respect to the knot sum operation.

Deciding if a particular knot is the unknot was a major driving force behind knot invariant
Knot invariant

In the mathematics field of knot theory, a knot invariant is a quantity defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism....
s, since it was thought this approach would possibly give an efficient algorithm to recognize the unknot
Unknotting problem

In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram....
 from some presentation such as a knot diagram.






Discussion
Ask a question about 'Unknot'
Start a new discussion about 'Unknot'
Answer questions from other users
Full Discussion Forum



Encyclopedia


The unknot arises in the mathematical theory of knots
Knot theory

In mathematics, knot theory is the area of topology that studies knot s. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs drastically in that the ends are joined together to prevent it from becoming undone....
. Intuitively, the unknot is a closed loop of rope without a knot
Knot

A knot is a method for fastening or securing linear material such as rope by tying or interweaving. It may consist of a length of one or more segments of rope, string, webbing, twine, strap, or even chain interwoven such that the line can bind to itself or to some other object?the "load"....
 in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the circle as a geometrically round circle. The unknot is also called the trivial knot. An unknot is the identity element
Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
 with respect to the knot sum operation.

Deciding if a particular knot is the unknot was a major driving force behind knot invariant
Knot invariant

In the mathematics field of knot theory, a knot invariant is a quantity defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism....
s, since it was thought this approach would possibly give an efficient algorithm to recognize the unknot
Unknotting problem

In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram....
 from some presentation such as a knot diagram. Currently there are several well-known unknot recognition algorithms (not using invariants), but they are either known to be inefficient or have no efficient implementation. It is not known whether the current invariants, such as finite type invariant
Finite type invariant

In the knot theory, a finite type invariant is a knot invariant that can be extended to an invariant of certain singular knots that vanishes on singular knots with m + 1 singularities and does not vanish on some singular knot with 'm' singularities....
s, are a complete invariant of the unknot. Even if they were, the problem of computing them efficiently remains.

Many useful practical knots are actually the unknot, including all knots which can be tied in the bight
Bight (knot)

The term bight is used in knot tying to refer to any curved section, slack part, or loop between the two ends of a rope, Twine, or yarn. An important concept, the term is used extensively in the description of knots and the discourse of knotting and related subjects....
.

See also

  • Knot (mathematics)
    Knot (mathematics)

    In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations ....
  • Unknotting problem
    Unknotting problem

    In mathematics, the unknotting problem is the problem of algorithmically recognizing the unknot, given some representation of a knot, e.g., a knot diagram....
  • Unlink
    Unlink

    In the mathematics field of knot theory, the 'unlink' is a Link that is equivalent to finitely many disjoint circles in the plane.* An n-component link L ⊂ 'S'3 is an unlink if and only if there exists n disjointly embedded discs Di ⊂ 'S'3 such that L...