Author: Eggtimer

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. It is then said to be of type or order m.

Goussarov, and (independently) Joan Birman and Xiao-Song Lin derived the following combinatorial definition of finite type invariant: Let V be a knot invariant. Define V¹ to be defined on a knot with one transverse singularity.

Consider a knot K to be a smooth embedding of a circle into ${\displaystyle \mathbb {R} ^{3}}$. Let K’ be a smooth immersion of a circle into ${\displaystyle \mathbb {R} ^{3}}$ with one transverse double point. Then

${\displaystyle V^{1}(K’)=V(K_{+})-V(K_{-})}$,

where ${\displaystyle K_{+}}$ is obtained from K by resolving the double point by pushing up one strand above the other, and ${\displaystyle K_{-}}$ is obtained similarly by pushing the opposite strand above the other. This can be done for maps with two transverse double points, three transverse double points, etc., by using the above relation. For V to be of finite type means precisely that there must be a positive integer m such that V vanishes on maps with ${\displaystyle m+1}$ transverse double points.

Furthermore, there is a notion of equivalence of knots with singularities being transverse double points such thatV respects this equivalence. There is also a notion of finite type invariant for 3-manifolds.

The simplest nontrivial Vassiliev invariant of knots is given by the coefficient of the quadratic term of the Alexander–Conway polynomial. It is an invariant of order two. Modulo two, it is equal to the Arf invariant.

Any coefficient of the Kontsevich invariant is a finite type invariant.

The Milnor invariants are finite type invariants of string links.^[1]

Michael Polyak and Oleg Viro gave a description of the first nontrivial invariants of orders 2 and 3 by means of Gauss diagram representations. Mikhail N. Goussarov has proved that all Vassiliev invariants can be represented that way.

In 1993, Maxim Kontsevich proved the following important theorem about Vassiliev invariants: For every knot one can compute an integral, now called the Kontsevich integral, which is a universal Vassiliev invariant, meaning that every Vassiliev invariant can be obtained from it by an appropriate evaluation. It is not known at present whether the Kontsevich integral, or the totality of Vassiliev invariants, is a complete knot invariant, or even if it detects the unknot. Computation of the Kontsevich integral, which has values in an algebra of chord diagrams, turns out to be rather difficult and has been done only for a few classes of knots up to now. There is no finite-type invariant of degree less than 11 which distinguishes mutant knots.^[2]

• Willerton’s fish

• Victor A. Vassiliev, Cohomology of knot spaces. Theory of singularities and its applications, 23–69, Adv. Soviet Math., 1, American Mathematical Society, Providence, RI, 1990.
• Joan Birman and Xiao-Song Lin, Knot polynomials and Vassiliev’s invariants. Inventiones Mathematicae, 111, 225–270 (1993)
• Bar-Natan, Dror (1995). “On the Vassiliev knot invariants”. Topology. 34 (2): 423–472. CiteSeerX 10.1.1.511.6301. doi:10.1016/0040-9383(95)93237-2.

• Weisstein, Eric W. “Vassiliev Invariant”. MathWorld.
• “Finite Type (Vassiliev) Invariants“, The Knot Atlas.

Trilene knot
Category	Hitch
Efficiency	85%
Typical use	Attaching a fishing line to a hook or swivel

The Trilene knot /ˈtraɪliːn/ is a multipurpose fishing knot that can be used for attaching monofilament line to hooks, swivels and lures. It resists slippage and failures.^[1] The knot was apparently in use at least as early as 1975 when it was included in Tom McNally’s Complete Book of Fishermen’s Knots as the “double-looped clinch knot”.^[2] However, professional anglers Jimmy Houston and Ricky Green would later claim that they invented the knot in the late 1970s while experimenting during promotional events for Trilene, a fishing line manufacturer. Both men favored the idea of naming the knot after themselves, though Trilene ultimately applied its own name instead.^[3] It is unclear whether Houston, Green or Trilene were aware of the knot’s earlier invention or its prior inclusion in McNally’s book.

• Video instructions for tying the Trilene Knot
• Drawings on how to tie a trilene knot

Fingerloop braiding is a technique of making sturdy and decorative cords from threads. It is a type of braiding known as loop manipulation. The braid is made from loops of thread, attached at a central point, and the loops placed over the fingers and interlaced in different ways.^[1]

In Europe it originated in the Middle Ages, and excavations from London have produced numerous examples in silk from between the second half of the 12th century and first half of the 15th.^[2] From the 15th century onwards, various directions and recipes for different fingerloop braid techniques began to appear in books and in print.^[1]

A related technique, which involved the loops being placed over the hand or fingers, is the Japanese kute-uchi style.^[3] This technique arose in the 7th Century, and were used through the Middle Ages to the 19th century, for uses such as tying armour.^[4]

Fingerloop braids were a type of braided cord with many uses. Beginning in the 13th century, they were used for lacing up clothing for a tighter fit. They were used to hold up men’s hose and to lace shoes. Braids were used to gather and tighten fabric at the neck and wrists of undergarments. Decorative cords were used to cinch purses in the same way.^[5]

Some wide and flat braids were made to be purely decorative and sewn on garments as trim.^[5]

Silk was a popular choice for fingerloop braids, both for its strength and its ability to be dyed many different colors. Leather was another popular material, especially for lacing shoes and tying armor. There is evidence that wool was used. Linen and flax were likely used, but little of those materials has survived.^[5]

• Kumihimo

• Online version of Compleat Anachronist #108: Fingerloop Braids
• Loop-Manipulation Braiding Research & Information Center (L–M BRIC) News
• Fingerloop Braiding: Gothic secrets and modern delights Archived 2012-02-24 at the Wayback Machine
• Sample fingerlooped braids from a fifteenth-century manuscript
• Brief history and instructions on Phiala’s String Page
• Loop Braiding

Trident loop
Category	Loop
Related	Ashley’s bend, Figure-eight knot, Zeppelin loop
Typical use	Forming fixed loop in end of a line

The trident loop is a fixed loop knot which can jam when heavily loaded. It was proposed as a replacement for the figure-of-eight loop for use in climbing by Robert M. Wolfe, MD, who developed it as a loop form of Ashley’s bend. While some tests indicate its strength lies somewhere between the weaker Bowline and stronger figure-of-eight loop, the trident loop shows exceptional resistance to slipping in shock-loading tests.^[1]

• 
  1. Start with a rope end.
• 
  2. Start an overhand knot, leaving enough rope for the loop and the rest of the knot.
• 
  3. Complete the overhand knot.
• 
  4. Form the loop by wrapping the working end around, and then form a bight in the working end.
• 
  5. Feed the bight through the overhand knot.
• 
  6. Wrap the remaining working end around the back of the knot.
• 
  7. Feed the working end up through the bight.
• 
  8. Tighten. This is the completed loop.

• List of knots

Figure-of-nine loop
Names	Figure-of-nine loop, Figure-nine loop
Category	Loop
Related	Figure-eight knot, Figure-of-eight follow through, Figure-of-eight loop, Stevedore knot
Typical use	Caving

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 extra half-turn before finishing the knot.^[1]

Also similar to the stevedore loop, the figure-nine loop is generally shown as being based on an intermediate form between the figure-eight knot and the stevedore knot.^[1][2] The Ashley Book of Knots shows this intermediate knot, in stopper form, as #521.^[3]

While it uses more rope and is bulkier than the figure-of-eight loop, the figure-nine loop is somewhat stronger and less likely to jam.^[1] It is sometimes used instead of a figure-of-eight loop to attach a rope to an anchor point or belay.^[2]

The knot can also be tied with the end of a rope – a single strand replaces the double strand, and therefore a naked end replaces the loop. This knot can be rearranged to form a stopper knot, in the same manner as a figure-of-eight stopper knot.

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 is an isotopy invariant, and hence can be used to distinguish between two different (non-isotopic) knots. In particular, since the unknot is not tricolorable, any tricolorable knot is necessarily nontrivial.

In these rules a strand in a knot diagram will be a piece of the string that goes from one undercrossing to the next.^[1] A knot is tricolorable if each strand of the knot diagram can be colored one of three colors, subject to the following rules:^[2]

1. At least two colors must be used, and

2. At each crossing, the three incident strands are either all the same color or all different colors.

Some references state instead that all three colors must be used.^[3] For a knot, this is equivalent to the definition above; however, for a link it is not.

“The trefoil knot and trivial 2-link are tricolorable, but the unknot, Whitehead link, and figure-eight knot are not. If the projection of a knot is tricolorable, then Reidemeister moves on the knot preserve tricolorability, so either every projection of a knot is tricolorable or none is.”^[2]

Here is an example of how to color a knot in accordance of the rules of tricolorability. By convention, knot theorists use the colors red, green, and blue.

Tricolorability is an isotopy invariant, which is a property of a knot or link that remains constant regardless of any ambient isotopy. This can be proven for tame knots by examining Reidemeister moves. Since each Reidemeister move can be made without affecting tricolorability, tricolorability is an isotopy invariant of tame knots.^[5]

Reidemeister Move I is tricolorable.	Reidemeister Move II is tricolorable.	Reidemeister Move III is tricolorable.

Because tricolorability is a binary classification (a link is either tricolorable or not*), it is a relatively weak invariant. The composition of a tricolorable knot with another knot is always tricolorable. A way to strengthen the invariant is to count the number of possible 3-colorings. In this case, the rule that at least two colors are used is relaxed and now every link has at least three 3-colorings (just color every arc the same color). In this case, a link is 3-colorable if it has more than three 3-colorings.

Any separable link with a tricolorable separable component is also tricolorable.

• Fox n-coloring
• Graph coloring

• Weisstein, Eric W. “Three-Colorable Knot”. MathWorld. Accessed: May 5, 2013.

A tressoir is a braid made of golden silk embroidered with metal and gems. It was worn by women in the 12th century.^[1][2]

Figure-eight loop
Names	Figure-eight loop, Flemish loop
Category	Loop
Related	figure-eight knot, flemish bend, Figure-of-nine loop, spider hitch
Releasing	Jamming
Typical use	climbing, caving
Caveat	jams
ABoK	#1047, #531
Instructions

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 created by a loop on the bight. It is used in climbing and caving.

The Flemish loop or figure-eight loop is perhaps stronger than the loop knot. Neither of these knots is used at sea, as they are hard to untie. In hooking a tackle to any of the loops, if the loop is long enough it is better to arrange the rope as a cat’s paw.

The double figure eight is used to put a loop in the end of a rope, or around an object. It is relatively easy to tie and is secure, but can become difficult to untie after heavy loading, and can jam badly in any rope type.

• Directional figure-eight knot
• List of knots

• Grog. “Figure Eight Follow Through”. Animated Knots. Retrieved 2018-07-12.
• Photographs showing how to tie the figure of 8 loop knot
• Notable Knot Index – Figure 8 Loop
• Video of a figure-eight loop being stressed until breaking, showing that stopper knot is not needed.

Trefoil
Common name	Overhand knot
Arf invariant	1
Braid length	3
Braid no.	2
Bridge no.	2
Crosscap no.	1
Crossing no.	3
Genus	1
Hyperbolic volume	0
Stick no.	6
Tunnel no.	1
Unknotting no.	1
Conway notation	[3]
A–B notation	31
Dowker notation	4, 6, 2
Last / Next	01 / 41
Other
alternating, torus, fibered, pretzel, prime, knot slice, reversible, tricolorable, twist

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 loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory.

The trefoil knot is named after the three-leaf clover (or trefoil) plant.

The trefoil knot can be defined as the curve obtained from the following parametric equations:

${\displaystyle {\begin{aligned}x&=\sin t+2\sin 2t\\y&=\cos t-2\cos 2t\\z&=-\sin 3t\end{aligned}}}$

The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus ${\displaystyle (r-2)^{2}+z^{2}=1}$:

${\displaystyle {\begin{aligned}x&=(2+\cos 3t)\cos 2t\\y&=(2+\cos 3t)\sin 2t\\z&=\sin 3t\end{aligned}}}$

Any continuous deformation of the curve above is also considered a trefoil knot. Specifically, any curve isotopic to a trefoil knot is also considered to be a trefoil. In addition, the mirror image of a trefoil knot is also considered to be a trefoil. In topology and knot theory, the trefoil is usually defined using a knot diagram instead of an explicit parametric equation.

In algebraic geometry, the trefoil can also be obtained as the intersection in C² of the unit 3-sphere S³ with the complex plane curve of zeroes of the complex polynomial z² + w³ (a cuspidal cubic).

A left-handed trefoil and a right-handed trefoil

If one end of a tape or belt is turned over three times and then pasted to the other, the edge forms a trefoil knot.^[1]

The trefoil knot is chiral, in the sense that a trefoil knot can be distinguished from its own mirror image. The two resulting variants are known as the left-handed trefoil and the right-handed trefoil. It is not possible to deform a left-handed trefoil continuously into a right-handed trefoil, or vice versa. (That is, the two trefoils are not ambient isotopic.)

Though chiral, the trefoil knot is also invertible, meaning that there is no distinction between a counterclockwise-oriented and a clockwise-oriented trefoil. That is, the chirality of a trefoil depends only on the over and under crossings, not the orientation of the curve.

But the knot has rotational symmetry. The axis is about a line perpendicular to the page for the 3-coloured image.

The trefoil knot is nontrivial, meaning that it is not possible to “untie” a trefoil knot in three dimensions without cutting it. Mathematically, this means that a trefoil knot is not isotopic to the unknot. In particular, there is no sequence of Reidemeister moves that will untie a trefoil.

Proving this requires the construction of a knot invariant that distinguishes the trefoil from the unknot. The simplest such invariant is tricolorability: the trefoil is tricolorable, but the unknot is not. In addition, virtually every major knot polynomial distinguishes the trefoil from an unknot, as do most other strong knot invariants.

In knot theory, the trefoil is the first nontrivial knot, and is the only knot with crossing number three. It is a prime knot, and is listed as 3₁ in the Alexander-Briggs notation. The Dowker notation for the trefoil is 4 6 2, and the Conway notation is [3].

The trefoil can be described as the (2,3)-torus knot. It is also the knot obtained by closing the braid σ₁³.

The trefoil is an alternating knot. However, it is not a slice knot, meaning it does not bound a smooth 2-dimensional disk in the 4-dimensional ball; one way to prove this is to note that its signature is not zero. Another proof is that its Alexander polynomial does not satisfy the Fox-Milnor condition.

The trefoil is a fibered knot, meaning that its complement in ${\displaystyle S^{3}}$ is a fiber bundle over the circle ${\displaystyle S^{1}}$. The trefoil K may be viewed as the set of pairs ${\displaystyle (z,w)}$ of complex numbers such that ${\displaystyle |z|^{2}+|w|^{2}=1}$ and ${\displaystyle z^{2}+w^{3}=0}$. Then this fiber bundle has the Milnor map ${\displaystyle \phi (z,w)=(z^{2}+w^{3})/|z^{2}+w^{3}|}$ as the fibre bundle projection of the knot complement ${\displaystyle S^{3}\setminus \mathbf {K} }$ to the circle ${\displaystyle S^{1}}$. The fibre is a once-punctured torus. Since the knot complement is also a Seifert fibred with boundary, it has a horizontal incompressible surface—this is also the fiber of the Milnor map. (This assumes the knot has been thickened to become a solid torus N_ε(K), and that the interior of this solid torus has been removed to create a compact knot complement ${\displaystyle S^{3}\setminus \operatorname {int} (\mathrm {N} _{\varepsilon }(\mathbf {K} )}$.)

The Alexander polynomial of the trefoil knot is$${\displaystyle \Delta (t)=t-1+t^{-1},}$$since ${\displaystyle {\begin{pmatrix}1&-1\\0&1\end{pmatrix}}}$ is a possible Seifert matrix (for the left-hand one), or because of its Conway polynomial,^[2] which is$${\displaystyle \nabla (z)=z^{2}+1.}$$The Jones polynomial is$${\displaystyle V(q)=q^{-1}+q^{-3}-q^{-4},}$$and the Kauffman polynomial of the trefoil is$${\displaystyle L(a,z)=za^{5}+z^{2}a^{4}-a^{4}+za^{3}+z^{2}a^{2}-2a^{2}.}$$The HOMFLY polynomial of the trefoil is$${\displaystyle L(\alpha ,z)=-\alpha ^{4}+\alpha ^{2}z^{2}+2\alpha ^{2}.}$$

The knot group of the trefoil is given by the presentation$${\displaystyle \langle x,y\mid x^{2}=y^{3}\rangle ,}$$or equivalently^[3]$${\displaystyle \langle x,y\mid xyx=yxy\rangle .}$$This group is isomorphic to the braid group with three strands.

As the simplest nontrivial knot, the trefoil is a common motif in iconography and the visual arts. For example, the common form of the triquetra symbol is a trefoil, as are some versions of the Germanic Valknut.

In modern art, the woodcut Knots by M. C. Escher (1965) depicts three trefoil knots whose solid forms are twisted in different ways.^[4]

• Cinquefoil knot
• Figure-eight knot (mathematics)
• Gordian Knot
• Pretzel link
• Trefoil knot metathesis
• Triquetra symbol

• Wolframalpha: (2,3)-torus knot
• Trefoil knot 3d model www.cgtrader.com

Category:Topology

Figure-eight knot
Common name	Figure-eight knot
Arf invariant	1
Braid length	4
Braid no.	3
Bridge no.	2
Crosscap no.	2
Crossing no.	4
Genus	1
Hyperbolic volume	2.02988
Stick no.	7
Unknotting no.	1
Conway notation	[22]
A–B notation	41
Dowker notation	4, 6, 8, 2
Last / Next	31 / 51
Other
alternating, hyperbolic, fibered, prime, fully amphichiral, twist

In knot theory, a figure-eight knot (also called Listing’s knot^[1]) is the unique knot with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the
trefoil knot. The figure-eight knot is a prime knot.

The name is given because tying a normal figure-eight knot in a rope and then joining the ends together, in the most natural way, gives a model of the mathematical knot.

A simple parametric representation of the figure-eight knot is as the set of all points (x,y,z) where

${\displaystyle {\begin{aligned}x&=2\cos {(3t)}+\cos {(t)},\\y&=2\sin {(3t)}+\sin {(t)},\\z&=\sin {(4t)},\end{aligned}}}$

for t varying over the real numbers (see 2D visual realization at bottom right).

The figure-eight knot is prime, alternating, rational with an associated value
of 5/3,^[2] and is achiral. The figure-eight knot is also a fibered knot. This follows from other, less simple (but very interesting) representations of the knot:

(1) It is a homogeneous^{[note 1]} closed braid (namely, the closure of the 3-string braid σ₁σ₂⁻¹σ₁σ₂⁻¹), and a theorem of John Stallings shows that any closed homogeneous braid is fibered.

(2) It is the link at (0,0,0,0) of an isolated critical point of a real-polynomial map F: R⁴→R², so (according to a theorem of John Milnor) the Milnor map of F is actually a fibration. Bernard Perron found the first such F for this knot, namely,

${\displaystyle F(x,y,z,t)=G(x,y,z^{2}-t^{2},2zt),\,\!}$

where

${\displaystyle {\begin{aligned}G(x,y,z,t)=\ &(z(x^{2}+y^{2}+z^{2}+t^{2})+x(6x^{2}-2y^{2}-2z^{2}-2t^{2}),\\&\ tx{\sqrt {2}}+y(6x^{2}-2y^{2}-2z^{2}-2t^{2})).\end{aligned}}}$

The figure-eight knot has played an important role historically (and continues to do so) in the theory of 3-manifolds. Sometime in the mid-to-late 1970s, William Thurston showed that the figure-eight was hyperbolic, by decomposing its complement into two ideal hyperbolic tetrahedra. (Robert Riley and Troels Jørgensen, working independently of each other, had earlier shown that the figure-eight knot was hyperbolic by other means.) This construction, new at the time, led him to many powerful results and methods. For example, he was able to show that all but ten Dehn surgeries on the figure-eight knot resulted in non-Haken, non-Seifert-fibered irreducible 3-manifolds; these were the first such examples. Many more have been discovered by generalizing Thurston’s construction to other knots and links.

The figure-eight knot is also the hyperbolic knot whose complement has the smallest possible volume, ${\displaystyle 6\Lambda (\pi /3)\approx 2.02988…}$ (sequence A091518 in the OEIS), where ${\displaystyle \Lambda }$ is the Lobachevsky function.^[3] From this perspective, the figure-eight knot can be considered the simplest hyperbolic knot. The figure eight knot complement is a double-cover of the Gieseking manifold, which has the smallest volume among non-compact hyperbolic 3-manifolds.

The figure-eight knot and the (−2,3,7) pretzel knot are the only two hyperbolic knots known to have more than 6 exceptional surgeries, Dehn surgeries resulting in a non-hyperbolic 3-manifold; they have 10 and 7, respectively. A theorem of Lackenby and Meyerhoff, whose proof relies on the geometrization conjecture and computer assistance, holds that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot. However, it is not currently known whether the figure-eight knot is the only one that achieves the bound of 10. A well-known conjecture is that the bound (except for the two knots mentioned) is 6.

The figure-eight knot has genus 1 and is fibered.
Therefore its complement fibers over the circle, the fibers being Seifert surfaces which are 2-dimensional tori with one boundary component.
The monodromy map is then a homeomorphism of the 2-torus, which can be represented in this case by the matrix ${\displaystyle ({\begin{smallmatrix}2&1\\1&1\end{smallmatrix}})}$.

The Alexander polynomial of the figure-eight knot is

${\displaystyle \Delta (t)=-t+3-t^{-1},\ }$

since ${\displaystyle {\begin{pmatrix}1&-1\\0&-1\end{pmatrix}}}$
is a possible Seifert matrix, or because of its Conway polynomial, which is

${\displaystyle \nabla (z)=1-z^{2},\ }$^[4]

and the Jones polynomial is

${\displaystyle V(q)=q^{2}-q+1-q^{-1}+q^{-2}.\ }$

The symmetry between ${\displaystyle q}$ and ${\displaystyle q^{-1}}$ in the Jones polynomial reflects the fact that the figure-eight knot is achiral.

• Ian Agol, Bounds on exceptional Dehn filling, Geometry & Topology 4 (2000), 431–449. MR 1799796
• Chun Cao and Robert Meyerhoff, The orientable cusped hyperbolic 3-manifolds of minimum volume, Inventiones Mathematicae, 146 (2001), no. 3, 451–478. MR 1869847
• Marc Lackenby, Word hyperbolic Dehn surgery, Inventiones Mathematicae 140 (2000), no. 2, 243–282. MR 1756996
• Marc Lackenby and Robert Meyerhoff, The maximal number of exceptional Dehn surgeries, arXiv:0808.1176
• Robion Kirby, Problems in low-dimensional topology, (see problem 1.77, due to Cameron Gordon, for exceptional slopes)
• William Thurston, The Geometry and Topology of Three-Manifolds, Princeton University lecture notes (1978–1981).

• “4_1“, The Knot Atlas. Accessed: 7 May 2013.
• Weisstein, Eric W. “Figure Eight Knot”. MathWorld.