## Table of Contents

34 relations: Alexander polynomial, Ambient isotopy, Bight (knot), Circle, Co-NP, Connected sum, Conway knot, Crossing number (knot theory), Cyclic group, Disk (mathematics), Embedding, Finite type invariant, Floer homology, Homeomorphism, Identity element, Jones polynomial, Khovanov homology, Kinoshita–Terasaka knot, Knot (mathematics), Knot complement, Knot group, Knot invariant, Knot theory, Linkage (mechanical), Morwen Thistlethwaite, NP (complexity), Seifert surface, Solid torus, Sphere, Stick number, Stuck unknot, Unknotting problem, Wild knot, 3-sphere.

- Fibered knots and links
- Non-alternating knots and links
- Non-tricolorable knots and links
- Prime knots and links
- Slice knots and links
- Torus knots and links

## Alexander polynomial

In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. Unknot and Alexander polynomial are knot theory.

See Unknot and Alexander polynomial

## Ambient isotopy

In the mathematical subject of topology, an ambient isotopy, also called an h-isotopy, is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold.

See Unknot and Ambient isotopy

## Bight (knot)

In knot tying, a bight is a curved section or slack part between the two ends of a rope, string, or yarn.

## Circle

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Unknot and circle are circles.

## Co-NP

In computational complexity theory, co-NP is a complexity class.

See Unknot and Co-NP

## Connected sum

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Unknot and connected sum are knot theory.

## Conway knot

In mathematics, in particular in knot theory, the Conway knot (or Conway's knot) is a particular knot with 11 crossings, named after John Horton Conway. Unknot and Conway knot are knot theory, non-alternating knots and links, non-tricolorable knots and links, prime knots and links and Slice knots and links.

## Crossing number (knot theory)

In the mathematical area of knot theory, the crossing number of a knot is the smallest number of crossings of any diagram of the knot.

See Unknot and Crossing number (knot theory)

## Cyclic group

In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently \Zn or Zn, not to be confused with the commutative ring of p-adic numbers), that is generated by a single element.

## Disk (mathematics)

In geometry, a disk (also spelled disc). Unknot and disk (mathematics) are circles.

See Unknot and Disk (mathematics)

## Embedding

In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

## Finite type invariant

In the mathematical theory of knots, a finite type invariant, or Vassiliev invariant (so named after Victor Anatolyevich Vassiliev), is a knot invariant that can be extended (in a precise manner to be described) 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.

See Unknot and Finite type invariant

## Floer homology

In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology.

## Homeomorphism

In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function.

## Identity element

In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied.

See Unknot and Identity element

## Jones polynomial

In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Unknot and Jones polynomial are knot theory.

See Unknot and Jones polynomial

## Khovanov homology

In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex.

See Unknot and Khovanov homology

## Kinoshita–Terasaka knot

In knot theory, the Kinoshita–Terasaka knot is a particular prime knot. Unknot and Kinoshita–Terasaka knot are knot theory, non-alternating knots and links, non-tricolorable knots and links, prime knots and links and Slice knots and links.

See Unknot and Kinoshita–Terasaka knot

## Knot (mathematics)

In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of which takes one knot to the other.

See Unknot and Knot (mathematics)

## Knot complement

In mathematics, the knot complement of a tame knot K is the space where the knot is not. Unknot and knot complement are knot theory.

See Unknot and Knot complement

## Knot group

In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space.

## Knot invariant

In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots.

## Knot theory

In topology, knot theory is the study of mathematical knots.

## Linkage (mechanical)

A mechanical linkage is an assembly of systems connected to manage forces and movement.

See Unknot and Linkage (mechanical)

## Morwen Thistlethwaite

Morwen Bernard Thistlethwaite (born 5 June 1945) is a knot theorist and professor of mathematics for the University of Tennessee in Knoxville.

See Unknot and Morwen Thistlethwaite

## NP (complexity)

In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems.

See Unknot and NP (complexity)

## Seifert surface

In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link. Unknot and Seifert surface are knot theory.

See Unknot and Seifert surface

## Solid torus

In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle.

## Sphere

A sphere (from Greek) is a geometrical object that is a three-dimensional analogue to a two-dimensional circle.

## Stick number

In the mathematical theory of knots, the stick number is a knot invariant that intuitively gives the smallest number of straight "sticks" stuck end to end needed to form a knot.

## Stuck unknot

In mathematics, a stuck unknot is a closed polygonal chain in three-dimensional space (a skew polygon) that is topologically equal to the unknot but cannot be deformed to a simple polygon when interpreted as a mechanical linkage, by rigid length-preserving and non-self-intersecting motions of its segments.

## 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. Unknot and unknotting problem are knot theory.

See Unknot and Unknotting problem

## Wild knot

In the mathematical theory of knots, a knot is tame if it can be "thickened", that is, if there exists an extension to an embedding of the solid torus S^1\times D^2 into the 3-sphere.

## 3-sphere

In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere.

## See also

### Fibered knots and links

- (−2,3,7) pretzel knot
- 62 knot
- 63 knot
- 71 knot
- Carrick mat
- Cinquefoil knot
- Fibered knot
- Figure-eight knot (mathematics)
- Hopf link
- Perko pair
- Torus knot
- Trefoil knot
- Unknot

### Non-alternating knots and links

### Non-tricolorable knots and links

- (−2,3,7) pretzel knot
- 62 knot
- 63 knot
- 71 knot
- Borromean rings
- Carrick mat
- Cinquefoil knot
- Conway knot
- Figure-eight knot (mathematics)
- Hopf link
- Kinoshita–Terasaka knot
- L10a140 link
- Perko pair
- Solomon's knot
- Stevedore knot (mathematics)
- Three-twist knot
- Unknot
- Whitehead link

### Prime knots and links

- 62 knot
- 63 knot
- 71 knot
- 74 knot
- Carrick mat
- Cinquefoil knot
- Conway knot
- Figure-eight knot (mathematics)
- Hopf link
- Kinoshita–Terasaka knot
- List of prime knots
- Perko pair
- Prime knot
- Stevedore knot (mathematics)
- Three-twist knot
- Trefoil knot
- Unknot
- Whitehead link

### Slice knots and links

- Conway knot
- Kinoshita–Terasaka knot
- Ribbon knot
- Slice knot
- Square knot (mathematics)
- Stevedore knot (mathematics)
- Trefoil knot
- Unknot

### Torus knots and links

## References

Also known as 0 1 knot, 0₁ knot, Culprit unknot, Freedman unknot, Goerlitz unknot, Haken's Gordian unknot, Ligocki-Sethian unknot, Not knot, Ochiai unknot, Standard unknot, Thistlethwaite knot, Thistlethwaite unknot, Thistlethwaite's unknot, Trivial knot, Twisted Freedman unknot.