Author: Eggtimer

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 of a given knot.

The first knot polynomial, the Alexander polynomial, was introduced by James Waddell Alexander II in 1923. Other knot polynomials were not found until almost 60 years later.

In the 1960s, John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander–Conway polynomial. The significance of this skein relation was not realized until the early 1980s, when Vaughan Jones discovered the Jones polynomial. This led to the discovery of more knot polynomials, such as the so-called HOMFLY polynomial.

Soon after Jones’ discovery, Louis Kauffman noticed the Jones polynomial could be computed by means of a partition function (state-sum model), which involved the bracket polynomial, an invariant of framed knots. This opened up avenues of research linking knot theory and statistical mechanics.

In the late 1980s, two related breakthroughs were made. Edward Witten demonstrated that the Jones polynomial, and similar Jones-type invariants, had an interpretation in Chern–Simons theory. Viktor Vasilyev and Mikhail Goussarov started the theory of finite type invariants of knots. The coefficients of the previously named polynomials are known to be of finite type (after perhaps a suitable “change of variables”).

In recent years, the Alexander polynomial has been shown to be related to Floer homology. The graded Euler characteristic of the knot Floer homology of Peter Ozsváth and Zoltan Szabó is the Alexander polynomial.

Alexander–Briggs notation	Alexander polynomial ${\displaystyle \Delta (t)}$	Conway polynomial ${\displaystyle \nabla (z)}$	Jones polynomial ${\displaystyle V(q)}$	HOMFLY polynomial ${\displaystyle H(a,z)}$
${\displaystyle 0_{1}}$ (Unknot)	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
${\displaystyle 3_{1}}$ (Trefoil Knot)	${\displaystyle t-1+t^{-1}}$	${\displaystyle z^{2}+1}$	${\displaystyle q^{-1}+q^{-3}-q^{-4}}$	${\displaystyle -a^{4}+a^{2}z^{2}+2a^{2}}$
${\displaystyle 4_{1}}$ (Figure-eight Knot)	${\displaystyle -t+3-t^{-1}}$	${\displaystyle -z^{2}+1}$	${\displaystyle q^{2}-q+1-q^{-1}+q^{-2}}$	${\displaystyle a^{2}+a^{-2}-z^{2}-1}$
${\displaystyle 5_{1}}$ (Cinquefoil Knot)	${\displaystyle t^{2}-t+1-t^{-1}+t^{-2}}$	${\displaystyle z^{4}+3z^{2}+1}$	${\displaystyle q^{-2}+q^{-4}-q^{-5}+q^{-6}-q^{-7}}$	${\displaystyle -a^{6}z^{2}-2a^{6}+a^{4}z^{4}+4a^{4}z^{2}+3a^{4}}$
${\displaystyle 3_{1}\#3_{1}}$ (Granny Knot)	${\displaystyle \left(t-1+t^{-1}\right)^{2}}$	${\displaystyle \left(z^{2}+1\right)^{2}}$	${\displaystyle \left(q^{-1}+q^{-3}-q^{-4}\right)^{2}}$	${\displaystyle \left(-a^{4}+a^{2}z^{2}+2a^{2}\right)^{2}}$
${\displaystyle 3_{1}\#3_{1}^{*}}$ (Square Knot)	${\displaystyle \left(t-1+t^{-1}\right)^{2}}$	${\displaystyle \left(z^{2}+1\right)^{2}}$	${\displaystyle \left(q^{-1}+q^{-3}-q^{-4}\right)\left(q+q^{3}-q^{4}\right)}$	${\displaystyle \left(-a^{4}+a^{2}z^{2}+2a^{2}\right)\times }$ ${\displaystyle \left(-a^{-4}+a^{-2}z^{-2}+2a^{-2}\right)}$

Alexander–Briggs notation organizes knots by their crossing number.

Alexander polynomials and Conway polynomials can not recognize the difference of left-trefoil knot and right-trefoil knot, while the Jones polynomial can.

• 
  The left-trefoil knot.
• 
  The right-trefoil knot.

So we have the same situation as the granny knot and square knot since the addition of knots in ${\displaystyle \mathbb {R} ^{3}}$ is the product of knots in knot polynomials.

• Adams, Colin. The Knot Book. American Mathematical Society. ISBN 0-8050-7380-9.
• Lickorish, W. B. R. (1997). An Introduction to Knot Theory. Graduate Texts in Mathematics. Vol. 175. New York: Springer-Verlag. ISBN 0-387-98254-X.

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. The equivalence is often given by ambient isotopy but can be given by homeomorphism.^[1] Some invariants are indeed numbers (algebraic^[2]), but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory (for example, “a knot invariant is a rule that assigns to any knot K a quantity φ(K) such that if K and K’ are equivalent then φ(K) = φ(K’).”^[3]). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification,^[3][4] both in “enumeration” and “duplication removal”.^[2]

A knot invariant is a quantity defined on the set of all knots, which takes the same value for any two equivalent knots. For example, a knot group is a knot invariant.^[5]

Typically a knot invariant is a combinatorial quantity defined on knot diagrams. Thus if two knot diagrams differ with respect to some knot invariant, they must represent different knots. However, as is generally the case with topological invariants, if two knot diagrams share the same values with respect to a [single] knot invariant, then we still cannot conclude that the knots are the same.^[6]

From the modern perspective, it is natural to define a knot invariant from a knot diagram. Of course, it must be unchanged (that is to say, invariant) under the Reidemeister moves (“triangular moves”^[4]). Tricolorability (and n-colorability) is a particularly simple and common example. Other examples are knot polynomials, such as the Jones polynomial, which are currently among the most useful invariants for distinguishing knots from one another, though currently it is not known whether there exists a knot polynomial which distinguishes all knots from each other.^[7][8][9] However, there are invariants which distinguish the unknot from all other knots, such as Khovanov homology and knot Floer homology.

Other invariants can be defined by considering some integer-valued function of knot diagrams and taking its minimum value over all possible diagrams of a given knot. This category includes the crossing number, which is the minimum number of crossings for any diagram of the knot, and the bridge number, which is the minimum number of bridges for any diagram of the knot.

Historically, many of the early knot invariants are not defined by first selecting a diagram but defined intrinsically, which can make computing some of these invariants a challenge. For example, knot genus is particularly tricky to compute, but can be effective (for instance, in distinguishing mutants).

The complement of a knot itself (as a topological space) is known to be a “complete invariant” of the knot by the Gordon–Luecke theorem in the sense that it distinguishes the given knot from all other knots up to ambient isotopy and mirror image. Some invariants associated with the knot complement include the knot group which is just the fundamental group of the complement. The knot quandle is also a complete invariant in this sense but it is difficult to determine if two quandles are isomorphic. The peripheral subgroup can also work as a complete invariant.^[10]

By Mostow–Prasad rigidity, the hyperbolic structure on the complement of a hyperbolic link is unique, which means the hyperbolic volume is an invariant for these knots and links. Volume, and other hyperbolic invariants, have proven very effective, utilized in some of the extensive efforts at knot tabulation.

In recent years, there has been much interest in homological invariants of knots which categorify well-known invariants. Heegaard Floer homology is a homology theory whose Euler characteristic is the Alexander polynomial of the knot. It has been proven effective in deducing new results about the classical invariants. Along a different line of study, there is a combinatorially defined cohomology theory of knots called Khovanov homology whose Euler characteristic is the Jones polynomial. This has recently been shown to be useful in obtaining bounds on slice genus whose earlier proofs required gauge theory. Mikhail Khovanov and Lev Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants. Catharina Stroppel gave a representation theoretic interpretation of Khovanov homology by categorifying quantum group invariants.

There is also growing interest from both knot theorists and scientists in understanding “physical” or geometric properties of knots and relating it to topological invariants and knot type. An old result in this direction is the Fáry–Milnor theorem states that if the total curvature of a knot K in ${\displaystyle \mathbb {R} ^{3}}$ satisfies

${\displaystyle \oint _{K}\kappa \,ds\leq 4\pi ,}$

where κ(p) is the curvature at p, then K is an unknot. Therefore, for knotted curves,

${\displaystyle \oint _{K}\kappa \,ds>4\pi .\,}$

An example of a “physical” invariant is ropelength, which is the length of unit-diameter rope needed to realize a particular knot type.

• Linking number – How many times curves wind around each other
• Finite type invariant – Type of invariant in Knot theory (or Vassiliev or Vassiliev–Goussarov invariant)
• Stick number – Smallest number of edges of an equivalent polygonal path for a knot
• Arnold invariants – Mathematical invariants used to classify plane curves

• Rolfsen, Dale (2003). Knots and Links. Providence, RI: AMS. ISBN 0-8218-3436-3.
• Adams, Colin Conrad (2004). The Knot Book: an Elementary Introduction to the Mathematical Theory of Knots (Repr., with corr ed.). Providence, RI: AMS. ISBN 0-8218-3678-1.
• Burde, Gerhard; Zieschang, Heiner (2002). Knots (2nd rev. and extended ed.). New York: De Gruyter. ISBN 3-11-017005-1.

• Cha, Jae Choon; Livingston, Charles. “KnotInfo: Table of Knot Invariants”. Indiana.edu. Archived from the original on 19 October 2022. Retrieved 17 August 2021.
• “Invariants“, The Knot Atlas.

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 the knot complement of K in R³,

${\displaystyle \pi _{1}\left(\mathbb {R} ^{3}\setminus K\right).}$

Other conventions consider knots to be embedded in the 3-sphere, in which case the knot group is the fundamental group of its complement in ${\displaystyle S^{3}}$.

Two equivalent knots have isomorphic knot groups, so the knot group is a knot invariant and can be used to distinguish between certain pairs of inequivalent knots. This is because an equivalence between two knots is a self-homeomorphism of ${\displaystyle \mathbb {R} ^{3}}$ that is isotopic to the identity and sends the first knot onto the second. Such a homeomorphism restricts onto a homeomorphism of the complements of the knots, and this restricted homeomorphism induces an isomorphism of fundamental groups. However, it is possible for two inequivalent knots to have isomorphic knot groups (see below for an example).

The abelianization of a knot group is always isomorphic to the infinite cyclic group Z; this follows because the abelianization agrees with the first homology group, which can be easily computed.

The knot group (or fundamental group of an oriented link in general) can be computed in the Wirtinger presentation by a relatively simple algorithm.

• The unknot has knot group isomorphic to Z.
• The trefoil knot has knot group isomorphic to the braid group B₃. This group has the presentation

${\displaystyle \langle x,y\mid x^{2}=y^{3}\rangle }$ or ${\displaystyle \langle a,b\mid aba=bab\rangle .}$

• A (p,q)-torus knot has knot group with presentation

${\displaystyle \langle x,y\mid x^{p}=y^{q}\rangle .}$

• The figure eight knot has knot group with presentation

${\displaystyle \langle x,y\mid yxy^{-1}xy=xyx^{-1}yx\rangle }$

• The square knot and the granny knot have isomorphic knot groups, yet these two knots are not equivalent.

• Link group

• Hazewinkel, Michiel, ed. (2001), “Knot and Link Groups“, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104

The knot complement of the unknot is homeomorphic to a solid torus – notice that while the unknot itself can be represented as a torus, the hole in the unknot corresponds to the solid region of the complement, while the knot itself is the hole in the complement. This is connected to the trivial Heegaard decomposition of the 3-sphere into two solid tori.

In mathematics, the knot complement of a tame knot K is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that K is a knot in a three-manifold M (most often, M is the 3-sphere). Let N be a tubular neighborhood of K; so N is a solid torus. The knot complement is then the complement of N,

${\displaystyle X_{K}=M-{\mbox{interior}}(N).}$

The knot complement X_K is a compact 3-manifold; the boundary of X_K and the boundary of the neighborhood N are homeomorphic to a two-torus. Sometimes the ambient manifold M is understood to be the 3-sphere. Context is needed to determine the usage. There are analogous definitions for the link complement.

Many knot invariants, such as the knot group, are really invariants of the complement of the knot. When the ambient space is the three-sphere no information is lost: the Gordon–Luecke theorem states that a knot is determined by its complement. That is, if K and K′ are two knots with homeomorphic complements then there is a homeomorphism of the three-sphere taking one knot to the other.

Knot complements are Haken manifolds.^[1] More generally complements of links are Haken manifolds.

• Knot genus
• Seifert surface

• C. Gordon and J. Luecke, “Knots are determined by their Complements”, J. Amer. Math. Soc., 2 (1989), 371–415.

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 equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of R³ which takes one knot to the other.

A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed — there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term knot is also applied to embeddings of S^j in Sⁿ, especially in the case j = n − 2. The branch of mathematics that studies knots is known as knot theory and has many relations to graph theory.

A knot is an embedding of the circle (S¹) into three-dimensional Euclidean space (R³),^[1] or the 3-sphere (S³), since the 3-sphere is compact.^{[2][Note 1]} Two knots are defined to be equivalent if there is an ambient isotopy between them.^[3]

The simplest knot, called the unknot or trivial knot, is a round circle embedded in R³.^[4] In the ordinary sense of the word, the unknot is not “knotted” at all. The simplest nontrivial knots are the trefoil knot (3₁ in the table), the figure-eight knot (4₁) and the cinquefoil knot (5₁).^[5]

Several knots, linked or tangled together, are called links. Knots are links with a single component.

Tame vs. wild knots

A polygonal knot is a knot whose image in R³ is the union of a finite set of line segments.^[6] A tame knot is any knot equivalent to a polygonal knot.^{[6][Note 2]} Knots which are not tame are called wild,^[7] and can have pathological behavior.^[7] In knot theory and 3-manifold theory, often the adjective “tame” is omitted. Smooth knots, for example, are always tame.

Knot complement

Given a knot in the 3-sphere, the knot complement is all the points of the 3-sphere not contained in the knot. A major theorem of Gordon and Luecke states that at most two knots have homeomorphic complements (the original knot and its mirror reflection). This in effect turns the study of knots into the study of their complements, and in turn into 3-manifold theory.^[10]

In knot theory, the connected sum (or knot sum) ${\displaystyle K_{1}\#K_{2}}$ of two oriented knots is the natural way to combine them. It makes the set of oriented knot types into a commutative monoid with the unknot as identity, and the operation is central to the decomposition of knots into prime pieces.

A knot is prime if it is non‑trivial and not a connected sum of two non‑trivial knots. The fundamental theorem, proved by Horst Schubert in 1949, states:

Every non‑trivial oriented knot decomposes as a connected sum of prime knots. The decomposition is unique up to the order of the factors.

Thus the monoid of oriented knot types is a free commutative monoid on the set of prime knots. The unknot plays the role of the unit, analogous to the number 1. For links of more than one component, unique decomposition fails.

Many polynomial and homological invariants are multiplicative under the connected sum:

Invariant	Behaviour under ${\displaystyle K_{1}\#K_{2}}$
Alexander polynomial	${\displaystyle \Delta _{K_{1}\#K_{2}}(t)=\Delta _{K_{1}}(t)\,\Delta _{K_{2}}(t)}$
Jones polynomial	${\displaystyle V_{K_{1}\#K_{2}}(t)=V_{K_{1}}(t)\,V_{K_{2}}(t)}$
HOMFLY polynomial	${\displaystyle P_{K_{1}\#K_{2}}(\ell ,m)=P_{K_{1}}(\ell ,m)\,P_{K_{2}}(\ell ,m)}$
Knot Floer homology	${\displaystyle {\widehat {\mathit {HFK}}}(K_{1}\#K_{2})\cong {\widehat {\mathit {HFK}}}(K_{1})\otimes {\widehat {\mathit {HFK}}}(K_{2})}$ (with grading shifts)
Khovanov homology	${\displaystyle {\mathit {Kh}}(K_{1}\#K_{2})\cong {\mathit {Kh}}(K_{1})\otimes {\mathit {Kh}}(K_{2})}$ (over a field)

The multiplicativity of the Alexander, Jones, and HOMFLY polynomials follows immediately from the skein relations.

For ordered links, the connected sum can be performed on chosen components, but the result depends on the choice of components and the ordering. Unlike for knots, the monoid of links is not free and factorisation into prime links is not unique.

• Adams, C.C. (1994), The Knot Book, W.H. Freeman, ISBN 978-0716723934 {{citation}}: Check |isbn= value: checksum (help)
• Burde, G.; Zieschang, H. (2003), Knots (2nd ed.), De Gruyter, ISBN 3-11-017005-1
• Schubert, H. (1949), “Die eindeutige Zerlegbarkeit eines Knotens in Primknoten”, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl., 1949 (3): 57–104
• Lickorish, W.B.R. (1997), An Introduction to Knot Theory, GTM 175, Springer, ISBN 0-387-98254-X

In contemporary mathematics the term knot is sometimes used to describe a more general phenomenon related to embeddings. Given a manifold M with a submanifold N, one sometimes says N can be knotted in M if there exists an embedding of N in M which is not isotopic to N. Traditional knots form the case where N = S¹ and M = R³ or M = S³.^[19][20]

The Schoenflies theorem states that the circle does not knot in the 2-sphere: every topological circle in the 2-sphere is isotopic to a geometric circle.^[21] Alexander’s theorem states that the 2-sphere does not smoothly (or PL or tame topologically) knot in the 3-sphere.^[22] In the tame topological category, it’s known that the n-sphere does not knot in the n + 1-sphere for all n. This is a theorem of Morton Brown, Barry Mazur, and Marston Morse.^[23] The Alexander horned sphere is an example of a knotted 2-sphere in the 3-sphere which is not tame.^[24] In the smooth category, the n-sphere is known not to knot in the n + 1-sphere provided n ≠ 3. The case n = 3 is a long-outstanding problem closely related to the question: does the 4-ball admit an exotic smooth structure?

André Haefliger proved that there are no smooth j-dimensional knots in Sⁿ provided 2n − 3j − 3 > 0, and gave further examples of knotted spheres for all n > j ≥ 1 such that 2n − 3j − 3 = 0. n − j is called the codimension of the knot. An interesting aspect of Haefliger’s work is that the isotopy classes of embeddings of S^j in Sⁿ form a group, with group operation given by the connect sum, provided the co-dimension is greater than two. Haefliger based his work on Stephen Smale’s h-cobordism theorem. One of Smale’s theorems is that when one deals with knots in co-dimension greater than two, even inequivalent knots have diffeomorphic complements. This gives the subject a different flavour than co-dimension 2 knot theory. If one allows topological or PL-isotopies, Christopher Zeeman proved that spheres do not knot when the co-dimension is greater than 2. See a generalization to manifolds.

• Knot theory – Study of mathematical knots
• Knot invariant – Function of a knot that takes the same value for equivalent knots
• List of mathematical knots and links

• “Main_Page“, The Knot Atlas.
• The Manifold Atlas Project

A knot is an intentional complication in cordage^[2] which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a hitch fastens a rope to another object; a bend fastens two ends of a rope to each another; a loop knot is any knot creating a loop; and splice denotes any multi-strand knot, including bends and loops.^[3] A knot may also refer, in the strictest sense, to a stopper or knob at the end of a rope to keep that end from slipping through a grommet or eye.^[4] Knots have excited interest since ancient times for their practical uses, as well as their topological intricacy, studied in the area of topology known as knot theory.

Knots and knotting have been used and studied throughout history. For example, Chinese knotting is a decorative handicraft art that began as a form of Chinese folk art in the Tang and Song dynasty (960–1279 AD) in China, later popularized in the Ming. Knot theory is the recent mathematical study of knots.

Knots of ancient origin include the bottle sling, bowline, cat’s paw, clove hitch, cow hitch, double fisherman’s knot, eskimo bowline, figure-eight knot, half hitch, kalmyk loop, one-sided overhand bend, overhand knot, overhand loop, reef knot, running bowline, single hitch, thief knot, Turk’s head knot, and two half-hitches.

The eleven main knots of Chinese knotting are the four-flower knot, six-flower knot, Chinese button knot, double connection knot, double coin knot, agemaki, cross knot, square knot, Plafond knot, Pan Chang knot, and the good luck knot.

Knots of more recent origin include the friendship knot of Chinese knotting. The sheepshank knot originates from 1627^[5] while the Western Union splice originates from the beginning of telegraphy.^[6]

There is a large variety of knots, each with properties that make it suitable for a range of tasks. Some knots are used to attach the rope (or other knotting material) to other objects such as another rope, cleat, ring, or stake. Some knots are used to bind or constrict objects. Decorative knots usually bind to themselves to produce attractive patterns.

Teaching

While some people can look at diagrams or photos and tie the illustrated knots, others learn best by watching how a knot is tied. Knot tying skills are often transmitted by sailors, scouts, climbers, canyoners, cavers, arborists, rescue professionals, stagehands, fishermen, linemen and surgeons. The International Guild of Knot Tyers is an organization dedicated to the promotion of knot tying.

The list of knots is extensive, but common properties allow for a useful system of categorization. For example, loop knots share the attribute of having some kind of an anchor point constructed on the standing end (such as a loop or overhand knot) into which the working end is easily hitched, using a round turn. An example of this is the bowline. Constricting knots often rely on friction to cinch down tight on loose bundles; an example is the Miller’s knot. Knots may belong to more than one category.

Bend

A knot uniting two lines^[14] (for knots joining two ends of the same line, see binding knots or loops).

Binding

A knot that restricts object(s) by making multiple winds.

Coil knot

Knots used to tie up lines for storage.

Decorative knot

A complex knot exhibiting repeating patterns often constructed around and enhancing an object.

Hitch

A knot tied to a post, cable, ring, or spar.

Lashing

A knot used to hold (usually) poles together.

Loop

A knot used to create a closed circle in a line.

Plait (or braid)

A number of lines interwoven in a simple regular pattern.

Slip (or running)

A knot tied with a hitch around one of its parts. In contrast, a loop is closed with a bend. While a slip knot can be closed, a loop remains the same size.

Slipped

Some knots may be finished by passing a bight rather than the end, for ease of untying. The common shoelace knot is an example, being a reef knot with both ends slipped.

Seizing

A knot used to hold two lines or two parts of the same line together.^[14]

Sennit

A number of lines interwoven in a complex pattern. See also Chain sinnet.

Splice

A knot formed by interweaving strands of rope rather than whole lines. More time-consuming but usually stronger than simple knots.

Stopper

A knot tied to hold a line through a hole.

Whipping

A binding knot used to prevent another line from fraying.

Coxcombing is a decorative knotwork performed by sailors during the Age of Sail.

The general purpose was to dress-up, protect, or help identify specific items and parts of ships and boats.

It is still found today in some whippings and wrappings of small diameter line on boat tillers and ships’ wheels to enhance the grip, or to identify rudder amidships.

Knots used in coxcombing include Turk’s head knot, Flemish, French whipping, and others.

Knot theory is a branch of topology.^[17] It deals with the mathematical analysis of knots, their structure and properties, and with the relationships between different knots.^[17] However, it does not take into account the role of friction in knots; in fact, no satisfactory general theory of knots that takes into account friction exists.^[18]

In topology, a knot is a figure consisting of a single loop with any number of crossing or knotted elements: a closed curve in space which may be moved around so long as its strands never pass through each other. As a closed loop, a mathematical knot has no proper ends, and cannot be undone or untied; however, any physical knot in a piece of string can be thought of as a mathematical knot by fusing the two ends. A configuration of several knots winding around each other is called a link. Various mathematical techniques are used to classify and distinguish knots and links. For instance, the Alexander polynomial associates certain numbers with any given knot; these numbers are different for the trefoil knot, the figure-eight knot, and the unknot (a simple loop), showing that one cannot be moved into the other (without strands passing through each other).^[17]

A simple mathematical theory of hitches has been proposed by Bayman^[19] and extended by Maddocks and Keller.^[20] It makes predictions that are approximately correct when tested empirically.^[18] No similarly successful theory has been developed for knots in general.

Knot tying consists of the techniques and skills employed in tying a knot in rope, nylon webbing, or other articles. The proper tying of a knot can be the difference between an attractive knot and a messy one, and occasionally life and death. It is important to understand the often subtle differences between what works, and what does not. For example, many knots “spill” or pull through, particularly if they are not “backed up,” usually with a single or double overhand knot to make sure the end of the rope does not make its way through the main knot, causing all strength to be lost.

The hagfish is known to strip slime from its skin by tying itself into a simple overhand knot, and moving its body to make the knot travel toward the tail. It also uses this action in reverse (tail to head) to pry out flesh after biting into a carcass.^[21]

• Circuit topology
• Chinese knotting
• Gordian Knot
• International Guild of Knot Tyers
• List of knots
• Quipu

Klemheist knot
Names	Klemheist knot, French Machard knot
Category	Hitch
Related	Prusik knot, Bachmann knot, Blake’s hitch
Typical use	Rock climbing
ABoK	#1762
Instructions

The klemheist knot or French Machard knot is a type of friction hitch that grips the rope when weight is applied, and is free to move when the weight is released. It is used similarly to a Prusik knot or the Bachmann knot to ascend or descend a climbing rope. One advantage is that webbing can be used as an alternative to cord. The Klemheist is easier to slide up than a Prusik. The klemheist is also a way to attach a snubber to the anchor rope of small boats, with the advantage that it is easy to undo.^[1]

Sometimes the knot name is misspelled as “kleimheist”, with an extra “i”.

Make a secure loop of a cord (see Double fisherman’s knot) that is definitely narrower than the rope. Wrap from one end around the rope two or three times in a direction that will be down. The loose end is then threaded through the starting end, and carefully tightened to leave the wraps neat. In use, strain must be taken only on the hanging end. If the knot slips when load is placed on the hanging loop, re-tie around the climbing rope another time or two, until there is no slippage. Holding around the wraps will let you slide the knot up or down. Making the wraps slide when the hitch is under tension will create friction heat on the rope and the knot.

• List of knots
• List of friction hitch knots

• Grog. “Klemheist animated using rope”. Animated Knots.
• Chockstone: Friction Knots
• Mark Adams’s “Son of a Hitch: A Genealogy of Arborists’ Climbing Hitches”
• A scientific study of common friction knots Archived 2013-12-12 at the Wayback Machine

Killick hitch
Names	Killick hitch, Kelleg hitch, Timber Hitch and a Half Hitch
Category	Hitch
Related	Timber hitch
Typical use	Attach a rope to an oddly shaped object.
ABoK	#271, #1733, #2162

The killick hitch /ˈkɪlɪk/ is a type of hitch knot used to attach a rope to oddly shaped objects.^[1]: 32 It is a combination of a timber hitch tied in conjunction with a half hitch ^[1]: 23 which is added to lend support and stability when pulling or hoisting the object.^[2]

A killick is “a small anchor or weight for mooring a boat, sometimes consisting of a stone secured by pieces of wood”.^[3]

The killick hitch is used to anchor small boats, usually by using some odd shaped heavy object. It is used by oystermen because the anchor is more readily moved than with other methods.

• List of knots
• List of hitch knots

In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman.^[1] It is initially defined on a link diagram as

${\displaystyle F(K)(a,z)=a^{-w(K)}L(K)\,}$,

where ${\displaystyle w(K)}$ is the writhe of the link diagram and ${\displaystyle L(K)}$ is a polynomial in a and z defined on link diagrams by the following properties:

• ${\displaystyle L(O)=1}$ (O is the unknot).
• ${\displaystyle L(s_{r})=aL(s),\qquad L(s_{\ell })=a^{-1}L(s).}$
• L is unchanged under type II and III Reidemeister moves.

Here ${\displaystyle s}$ is a strand and ${\displaystyle s_{r}}$ (resp. ${\displaystyle s_{\ell }}$) is the same strand with a right-handed (resp. left-handed) curl added (using a type I Reidemeister move).

Additionally L must satisfy Kauffman’s skein relation:

The pictures represent the L polynomial of the diagrams which differ inside a disc as shown but are identical outside.

Kauffman showed that L exists and is a regular isotopy invariant of unoriented links. It follows easily that F is an ambient isotopy invariant of oriented links.

The Jones polynomial is a special case of the Kauffman polynomial, as the L polynomial specializes to the bracket polynomial. The Kauffman polynomial is related to Chern–Simons gauge theories for SO(N) in the same way that the HOMFLY polynomial is related to Chern–Simons gauge theories for SU(N).^[2]

• Kauffman, Louis (1987). On Knots. Annals of Mathematics Studies. Vol. 115. Princeton, NJ: Princeton University Press. ISBN 0-691-08435-1. MR 0907872.

• “Kauffman polynomial“, Encyclopedia of Mathematics
• “The Kauffman Polynomial“, The Knot Atlas.

Kalmyk Loop
Category	Loop
Origin	Ancient
Releasing	Quick Release
ABoK	Not Listed

The Kalmyk loop (Russian: калмыцкий узел) is a fixed loop still largely unused in the West, but common in Russia and often used instead of the bowline.

The knot is named after the Kalmyks, a nomad ethnicity in Russia.

It is very quick to tie, it is secure, and it undoes quickly when pulling the free end.
The knot is not mentioned in The Ashley Book of Knots but is found in its Russian equivalent, the book “Морские узлы” by Lev Skryagin.

Without the slip, the knot is known as the Eskimo bowline or Cossack knot.

• Скрягин Л. Н. Морские узлы — Москва, Транспорт, 1982

• russian article with pictures
• Kalmyk Loop (калмыцкий узел)
• Can the Kalmyk Unseat the Bowline as King of Knots?
• Little Campfires (2020-08-01). “How to Tie a Bowline Knot (Plus Variations): Kalmyk Loop Knot”. YouTube.
• TheTautLine (2023-09-11). “Complete Guide to the Bowline Knot and its Most Important Variants: Kalmyk Loop”. YouTube.
• Video: How to tie 1
• Video: How to tie 2
• Video: How to tie 3
• Video: 3 Ways to Tie the Kalmyk Loop (Including One You’ve Never Seen)