Research any topic before you write.

Find related topics. | Discover entities. | See connections. | Build a topical map.

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…

Standards, Examples & Invariants

Use the mouse wheel or two fingers (on touchscreens) to zoom in and out of the map.

Research this topic

Explore the main themes, entities and connections around Unknot. Start with the topic map, then use the sections below for research and deeper semantic analysis.

Explore this topic

Start with a few of the strongest sections from the source topic. These are research directions, not a list of keywords you must use.

Key facts & relationships

High-confidence facts extracted from structured source data. Use them as anchors for further research.

Arf invariant
0
A–B notation
01
Braid no.
1
Bridge no.
0
Common name
Circle
Conway notation
-

Topics to explore

Browse the full topic structure. Each item opens a new analysis centered on that subject.

Overview

Background

Unknotting problem

Examples

Invariants

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

Map overview Semantic statistics

Unknot

Nodes47
Edges46
Triples85
Avg. degree1.96
Density0.042553
Components1

How this topic connects Entity context

See the strongest relationship patterns around the current topic before diving into the raw triples.

Unknot

Top relations

related to Examples · 17
Unknot → Another, At, Culprit, Early, Goeritz, It, Ken Millett, Lebrecht Goeritz, Many, Monster, Note, Ochiai, Ochiai's, Reidemeister, Rob Scharein, Such, Thistlethwaite
is a · 8
Unknot → closed loop in three dimensions that does not contain knots and can, closed loop of rope without a knot tied into it, diagram of the unknot for which proving that it is unknotted is difficult, identity element with respect to the knot sum operation, only knot that is the boundary of an embedded disk, only knot whose knot group is an infinite cyclic group, particular unknotted linkage that cannot be reconfigured into a flat convex polygon, projection of its three dimensional shape onto two dimensions
related to background · 8
Unknot → An, At, II, III, Reidemeister, This, To, While
related to Invariants · 8
Unknot → Alexander, Conway, It, Jones, Kinoshita, No, Terasaka, The Alexander
related to Unknotting on a sphere · 8
Unknot → Computational, Culprit, Goeritz, If, In, Monster, North Pole, South Pole
related to External links · 6
Unknot → Accessed, Eric, MathWorld, May, The Knot Atlas, Weisstein
related to Unknotting problem · 6
Unknot → Deciding, Floer, It, Jones, Khovanov, NP
see also · 5
Unknot → Embedding, Euclidean, Knot, Link, Minimum
related to Hard unknot · 4
Unknot → Hard, Reidemeister, These, Typically
Arf invariant · 1
Unknot → 0

Important terminology Word statistics

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

knot diagram crossings hard crossing circle known unknots number unknotting reidemeister diagrams additional sphere moves many one least loop unknotted

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
UnknotArf invariant01.00infobox
UnknotA–B notation011.00infobox
UnknotBraid no.11.00infobox
UnknotBridge no.01.00infobox
UnknotCommon nameCircle1.00infobox
UnknotConway notation-1.00infobox
UnknotCrossing no.01.00infobox
UnknotDowker notation-1.00infobox
UnknotGenus01.00infobox
UnknotLinking no.01.00infobox
UnknotNext311.00infobox
UnknotStick no.31.00infobox
UnknotTunnel no.01.00infobox
UnknotUnknotting no.01.00infobox
Unknotis aclosed loop of rope without a knot tied into it0.90text
Unknotis aonly knot that is the boundary of an embedded disk0.90text
Unknotis aidentity element with respect to the knot sum operation0.90text
Unknotis aclosed loop in three dimensions that does not contain knots and can0.90text
Unknotis aprojection of its three dimensional shape onto two dimensions0.90text
Unknotis aparticular unknotted linkage that cannot be reconfigured into a flat convex polygon0.90text
Unknotis adiagram of the unknot for which proving that it is unknotted is difficult0.90text
Unknotis aonly knot whose knot group is an infinite cyclic group0.90text

Related concept clusters Concept neighborhoods

These clusters group vocabulary that occurs around closely connected concepts in the source material.

    Connections between topic areas Semantic bridges

    Bridges can reveal useful research angles that are easy to miss in a flat list of related terms.

    Min side: 3
    For writers, content strategists, SEOs, marketers and creators — from quick topic research to advanced semantic analysis.