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Unique knot with a crossing number of four
In knot theory, a figure-eight knot (also called Listing's knot) is the unique knot with a crossing number of four. This makes it the knot with the third-smallest
Figure-eight knot (mathematics)
Figure-eight_knot_(mathematics)
Type of stopper knot used in sailing and climbing
The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in sailing, rock climbing and caving as a method of stopping
Figure-eight_knot
Mathematical knot with crossing number 6
complement having a volume of approximately 3.16396. Figure-eight knot (mathematics) "6_1", The Knot Atlas. Weisstein, Eric W. "Stevedore's Knot". MathWorld.
Stevedore_knot_(mathematics)
Simplest non-trivial closed knot with three crossings
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining the two
Trefoil_knot
Topics referred to by the same term
the rose with two petals Figure 8, shape described by an analemma, a curve in astronomy Figure-eight knot (mathematics), in knot theory Lemniscate, various
Figure_8
Type of knot
Figure-eight loop (also figure-eight on a bight, figure-eight follow-through, figure-eight retrace, Flemish loop, or Flemish eight) is a type of knot
Figure-eight_loop
The overhand knot, for example, is also known as the thumb knot. The figure-eight knot is also known as the Savoy knot or the Flemish knot. Contents Top
List_of_knots
of a knot. Two classes of knots: torus knots and pretzel knots Cinquefoil knot also known as a (5, 2) torus knot. Figure-eight knot (mathematics) the
List_of_knot_theory_topics
knot/Trefoil knot - (2,3)-torus knot, the two loose ends of a common overhand knot joined together 41 knot/Figure-eight knot (mathematics) - a prime knot with
List of mathematical knots and links
List_of_mathematical_knots_and_links
Family of mathematical knots
In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. (That
Twist_knot
Method of fastening or securing linear material
fisherman's knot, eskimo bowline, figure-eight knot, half hitch, kalmyk loop, one-sided overhand bend, overhand knot, overhand loop, reef knot, running bowline
Knot
Type of mathematical knot
In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disk with only ribbon singularities. Intuitively, this
Ribbon_knot
Embedding of the circle in three dimensional Euclidean space
In mathematics, a knot is an embedding of the circle (S1) into three-dimensional Euclidean space, R3 (also known as E3). Often two knots are considered
Knot_(mathematics)
or otherwise. An example is the figure-eight loop knot, which can be tied in the bight, by tying a figure-eight knot using a bight instead of the end
List_of_knot_terminology
Type of knot used to join two lengths of rope
fisherman's knot, and the double figure-eight bend. Bends allow the combined ropes to support weight or transmit force. The common reef knot (square knot) is
Bend_(knot)
Type of mathematical knot
geometric topology, a branch of mathematics, the (−2, 3, 7) pretzel knot, sometimes called the Fintushel–Stern knot (after Ron Fintushel and Ronald J
(−2,3,7)_pretzel_knot
Study of mathematical knots
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope,
Knot_theory
Prime knot named for John Horton Conway
In mathematics, specifically in knot theory, the Conway knot (or Conway's knot) is a particular knot with 11 crossings, named after John Horton Conway
Conway_knot
Type of mathematical knot
mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is
Satellite_knot
topology topics. Knot (mathematics) Link (knot theory) Wild knots Examples of knots (and links) Unknot Trefoil knot Figure-eight knot (mathematics) Borromean
List of geometric topology topics
List_of_geometric_topology_topics
Type of knot
stevedore knot is a stopper knot, often tied near the end of a rope. It is more bulky and less prone to jamming than the closely related figure-eight knot. The
Stevedore_knot
Fundamental group of a knot complement
In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of
Knot_group
Curved section between two ends of a rope
rope. An overhand knot tied in the bight results in an overhand loop. A figure-eight knot tied in the bight results in a figure-eight loop. Ashley (1944)
Bight_(knot)
Type of knot
just a badly water-swollen figure eight stopper knot. The oysterman's stopper...It is a larger knot than the figure-eight, which has but one part around
Ashley's_stopper_knot
Mathematical knot with crossing number 5
In knot theory, the three-twist knot is the twist knot with three-half twists. It is listed as the 52 knot in the Alexander-Briggs notation, and is one
Three-twist_knot
Knot that forms a fixed thicker point to prevent unreeving
handle. Knots commonly used for this purpose are: Overhand knot Double overhand knot Figure-eight knot Stevedore knot Ashley's stopper knot The Chinese
Stopper_knot
Binding hitch knot
The constrictor knot is one of the most effective binding knots. Simple and secure, it is a harsh knot that can be difficult or impossible to untie once
Constrictor_knot
Type of knot
overhand knot, it assumes a regular pentagonal shape. List of knots Trefoil knot, the mathematical treatment of the overhand knot Double overhand knot Slip
Overhand_knot
Type of knot
The slip knot is a stopper knot which is easily undone by pulling the tail (working end). The slip knot is related to the running knot, which will release
Slip_knot
Dangerous knot
The offset figure-eight bend is a poor knot that has been implicated in the deaths of several rock climbers.[failed verification] The knot may capsize
Offset_figure-eight_bend
Two interlinked loops with five structural crossings
overlay of a circle and a figure-eight shaped loop. A common way of describing this knot is formed by overlaying a figure-eight shaped loop with another
Whitehead_link
Type of knot
figure-eight loop, (also known as a bunny ears, or a dog eared loop) is a type of knot that forms two parallel loops, and resembles the figure-eight loop
Double_figure-eight_loop
Knot that bounds an embedded disk in 4-space
A slice knot is a mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space. A knot K ⊂ S 3 {\displaystyle K\subset
Slice_knot
Type of knot in knot theory
Bridge number 2 In the mathematical field of knot theory, a 2-bridge knot is a knot which can be regular isotoped so that the natural height function given
2-bridge_knot
Integer-valued knot invariant; least number of crossings in a knot diagram
zero, the trefoil knot three and the figure-eight knot four. There are no other knots with a crossing number this low, and just two knots have crossing number
Crossing_number_(knot_theory)
Type of mathematical link
knot (the figure-eight knot) 52 knot (the three-twist knot) 61 knot (the stevedore knot) 62 knot 63 knot 74 knot 10 161 knot (the "Perko pair" knot)
Hyperbolic_link
Type of knot
known as a figure eight bend, a rewoven figure eight, is a knot for joining two ropes of roughly similar size. A loose figure-eight knot is tied in the
Flemish_bend
Type of knot
The directional figure eight (a.k.a. inline figure-eight loop) is a loop knot. It is a knot that can be made on the bight. The loop must only be loaded
Directional_figure_eight
Type of knot
make it a member of a class of similar knots known as butcher's knots. A lightly tightened figure-eight knot is formed around the standing part of the
Packer's_knot
Knot which lies on the surface of a torus in 3-dimensional space
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies
Torus_knot
Type of knot
The figure-of-nine loop is a type of knot to form a fixed loop in a rope. Tied in the bight, it is made similarly to a figure-of-eight loop but with an
Figure-of-nine_loop
Mathematical knot with crossing number 7
In mathematical knot theory, 74 is the name of a 7-crossing knot which can be visually depicted in a highly-symmetric form, and so appears in the symbolism
74_knot
Type of knot
The Siberian hitch (or Evenk knot) is a hitch knot used to attach a rope to an object. It is a type of slipped figure-eight noose. The hitch is known for
Siberian_hitch
Mathematical invariant of a knot or link
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant
Jones_polynomial
Knot that can't be tied in a string of constant diameter
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
Wild_knot
In mathematics, especially in the area of topology known as knot theory, an invertible knot is a knot that can be continuously deformed to itself, but
Invertible_knot
Type of knot
butterfly loop, artillery loop, figure-eight loop or directional figure-eight loop, or another of many suitable loop knots. If a fixed loop is used repeatedly
Trucker's_hitch
Property of an object that is not congruent to its mirror image
a chiral knot. For example, the unknot and the figure-eight knot are achiral, whereas the trefoil knot is chiral. Asymmetry Chiral polytope Chirality
Chirality_(mathematics)
Type of knot
into the knot. The figure-8 water knot (or figure-8 bend or Flemish bend) is based upon a figure-8 (or Flemish) knot instead of an overhand knot. It is
Water_knot
Connected sum of two trefoil knots with opposite chirality
composite knots. The square knot is the mathematical version of the common reef knot. The square knot can be constructed from two trefoil knots, one of
Square_knot_(mathematics)
Necktie knot
Windsor knot, sometimes referred to as a full Windsor (or misleadingly as a double Windsor) to distinguish it from the half-Windsor, is a knot used to
Windsor_knot
Andean record-keeping system using knotted cords
overhand knot with one or more additional turns; and figure-eight knots. The Aschers also identified a fourth, and less common, type of knot—a figure-eight knot
Quipu
Mathematical knot
In knot theory, a branch of mathematics, a knot or link K {\displaystyle K} in the 3-dimensional sphere S 3 {\displaystyle S^{3}} is called fibered or
Fibered_knot
schemes. Conway knot 11n34 Kinoshita–Terasaka knot 11n42 List of knots List of mathematical knots and links Knot tabulation (−2,3,7) pretzel knot Originally
List_of_prime_knots
Property in knot theory
In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules
Tricolorability
Type of knot
The hangman's knot or hangman's noose (also known as a collar during the Elizabethan era) is a knot most often associated with its use in hanging a person
Hangman's_knot
Normalized hyperbolic volume of the complement of a hyperbolic knot
In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete
Hyperbolic_volume
Knot that is not equivalent to its mirror image
In the mathematical field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image (when identical while reversed). An oriented
Chiral_knot
Binding knot
The boa knot is a modern binding knot invented by weaver Peter Collingwood in 1996. His intention was to develop a knot that would hold well when the
Boa_knot
Type of knot
The fisherman's knot is a knot—specifically a bend—that joins two lines. The double fisherman's knot and triple fisherman's knot are variations used in
Fisherman's_knot
American mathematician (1946–2012)
exhibiting the hyperbolic structure of the figure-eight knot complement. He showed that the figure-eight knot complement could be decomposed as the union
William_Thurston
Illustration of the Pythagorean theorem
pants, Figure of the Bride, theorem of the married women, and chase of the little married women. According to Swiss-American historian of mathematics Florian
Bride's_Chair
Knot used to join two ropes together
inhibits any likelihood of capsizing. The offset figure-eight bend, a similar knot using the figure-eight knot, has been used in the belief that its greater
Offset_overhand_bend
1991 mathematical film
Not Knot is a 16-minute film on the mathematics of knot theory and low-dimensional topology, centered on and titled after the concept of a knot complement
Not_Knot
Polynomials arising in knot theory
In the mathematical field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable
HOMFLY_polynomial
Branch of mathematics
Dept. of Mathematics. Zbl 0208.48701. Adams, Colin (2004). The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical
Topology
Rope loop and knot
at the end of a rope in which the knot tightens under load and can be loosened without untying the knot. The knot can be used to secure a rope to a post
Noose
Connected sum of two trefoil knots with same chirality
the square knot are the simplest of all composite knots. The granny knot is the mathematical version of the common granny knot. The granny knot can be constructed
Granny_knot_(mathematics)
Type of knot
The granny knot is a binding knot, used to secure a rope or line around an object. It is considered inferior to the reef knot (square knot), which it
Granny_knot
Type of knot
strong as the blood knot, similar to the reverse figure of eight and stronger than the fisherman's bend, sheet bend or reef knot". In October 1978, an
Hunter's_bend
Mathematical knot with crossing number 6
0\leq \phi \leq 2\pi } . Like the figure-eight knot, the 63 knot is fully amphichiral. This means that the 63 knot is amphichiral, meaning that it is
63_knot
Common binding knot
The reef knot, or square knot, is an ancient and simple binding knot used to secure a rope or line around an object. It is sometimes also referred to
Reef_knot
Type of knot
The clove hitch is an ancient type of knot, made of two successive single hitches tied around an object. It is most effectively used to secure a middle
Clove_hitch
Type of knot
The shoelace knot, or bow knot, is commonly used for tying shoelaces and bow ties. The shoelace knot is a doubly slipped reef knot formed by joining the
Shoelace_knot
Type of knot
A shank is a type of knot that is used to shorten a rope or take up slack, such as the sheepshank. The sheepshank knot is not stable. It will fall apart
Sheepshank
Simple knot used to form a fixed loop at the end of a rope
simplest way, the mathematical knot obtained is the so-called 6₂ knot. The sequence of necessary moves is depicted here. List of knots Karash double loop
Bowline
Knot that makes a pair of fixed-size loops in the middle of a rope
European cavers widely advocate the use of a figure eight twisted version of the bowline on a bight. This knot can be used to provide a toe hold in the middle
Bowline_on_a_bight
Type of knot
of a rope. Although it is a stable knot, it is often backed up with a stopper knot, such as a figure-of-eight knot, for safety. It is used for both ascending
Blake's_hitch
Conjecture in knot theory relating quantum invariants and hyperbolic geometry
In the branch of mathematics called knot theory, the volume conjecture is an open problem that relates quantum invariants of knots to the hyperbolic geometry
Volume_conjecture
Three linked but pairwise separated rings
work of knot theory to include the Borromean rings was a catalog of knots and links compiled in 1876 by Peter Tait. In recreational mathematics, the Borromean
Borromean_rings
Type of knot
A Prusik (/ˈprʌsɪk/ PRUSS-ik) is a friction hitch or knot used to attach a loop of cord around a rope, applied in climbing, canyoneering, mountaineering
Prusik
Type of knot used to join a rope to an object
common types of hitch knots include the clove hitch, the timber hitch, and the round turn and two half-hitches. A simple mathematical theory of hitches has
Hitch_(knot)
Knot invariant
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander
Alexander_polynomial
In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the
Alternating_knot
Motif with two doubly-interlinked loops
classified as a link, and is not a true knot according to the definitions of mathematical knot theory. The Solomon's knot consists of two closed loops, which
Solomon's_knot
Type of bend knot
A blood knot (barrel knot) is a bend knot most usefully employed for joining sections of monofilament nylon line while maintaining a high portion of the
Blood_knot
Method of tying a necktie
The small knot, also known as oriental knot, Kent knot, or simple knot, is the simplest method of tying a necktie. Unlike the Four-in-hand knot and Windsor
Small_knot
Loop seen as a trivial knot
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
Unknot
Function of a knot that takes the same value for equivalent knots
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. The
Knot_invariant
Type of knot
The surgeon's knot is a surgical knot and is a simple modification to the reef knot. It adds an extra twist when tying the first throw, forming a double
Surgeon's_knot
Class of knot used to add weight to the end of a rope to make it easier to throw
A heaving line knot is a family of knots which are used for adding weight to the end of a rope, to make the rope easier to throw. In nautical use, a heaving
Heaving_line_knot
1944 encyclopedia of knots by Clifford W. Ashley
The Ashley Book of Knots (ABoK) is an encyclopedia of knots written and illustrated by the American sailor and artist Clifford W. Ashley. First published
The_Ashley_Book_of_Knots
American mathematician (born 1956)
smallest cusped orientable hyperbolic 3-manifolds are precisely the figure-eight knot complement and its sibling manifold. Adams has investigated and defined
Colin_Adams_(mathematician)
Mathematical knot with crossing number 6
In knot theory, the 62 knot is one of three prime knots with crossing number six, the others being the stevedore knot and the 63 knot. This knot is sometimes
62_knot
In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties
Knot_polynomial
Type of knot
Chinese button knot is essentially a knife lanyard knot where the lanyard loop is shortened to a minimum, i.e. tightened to the knot itself. There emerges
Chinese_button_knot
Knot used in fishing
The Palomar knot (/ˈpæləmɑːr/ PAL-ə-mar) is a knot that is used for securing a fishing line to a fishing lure, hook, or swivel. It is strong and easy to
Palomar_knot
Loop knot often perceived as having better security than a bowline
dangerous slip knot. Additional safety is achieved by tying with a tail (see below). When finished, the working end forms a figure eight. Because of the
Yosemite_bowline
Type of planar curve with tree-like structure
only be shadows of the unknot. As knot diagrams, these represent connected sums of figure-eight curves. Each figure-eight is unknotted and their connected
Tree-like_curve
American mathematician
hyperbolic manifolds and profinite groups. He proved that the figure-eight knot is the only knot whose complement is an arithmetic hyperbolic 3-manifold. With
Alan_Reid_(mathematician)
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FIGURE EIGHT-KNOT-MATHEMATICS
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FIGURE EIGHT-KNOT-MATHEMATICS
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