Knot theory

study of mathematical knots

link of three loops; simplest Brunnian link

geometric property of an object which cannot be mapped to its mirror image by rotations and translations alone

simplest non-trivial closed knot with three crossings

ornament with two doubly-interlinked loops

mathematical concept in knot theory

right|150px|thumb|Two simple diagrams of the unknot In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic (that is, deformable) to a geometrically round circle, the standard unknot.

geometric modification on manifolds

prime link; simplest nontrivial link

knot which lies on the surface of a torus in 3-dimensional space

knot invariant

mathematical invariant of a knot or link

two-variable knot polynomial, generalizing the Jones and Alexander polynomials

unique knot with a crossing number of four

surface whose boundary is a knot or a link

In knot theory, there are several competing notions of the quantity writhe, or \operatorname{Wr}. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amount of "coiling" of a mathematical knot (or any closed simple curve) in three-dimensional space and assumes real numbers as values. In both cases, writhe is a geometric quantity, meaning that while deforming a curve (or diagram) in such a way that does not change its topology, one may still change its writhe.

two interlinked loops with five structural crossings

prime knot with 11 crossings named for John Horton Conway

polynomial invariant of framed links

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complement of a knot in three-sphere

mathematical knot with crossing number 6

connected sum of two trefoil knots with opposite chirality

area of mathematics that is a combination of algebraic number theory and topology

two-variable polynomial knot invariant; related to SO(N) Chern–Simons theory

In the mathematical field of knot theory, an unlink is a link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane.

torus knot with crossing number 5

Relation between triples of links differing by 1 crossing, used to define knot invariants

mathematical knot with crossing number 5

three-dimensional smooth curves with small total curvature must be unknotted

connected sum of two trefoil knots with same chirality

concepts in abstract algebra

type of mathematical link

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mathematical knot with crossing number 7

assignment problem in combinatorial mathematics

normalized hyperbolic volume of the complement of a hyperbolic knot

embedding a graph in 3D space with no cycles interlinked

concept in mathematical knot theory