Category: Knots

Knots in Washington is an international conference on knot theory and its ramifications held twice a year since 1995. The main organizers are Józef Przytycki, Alexander Shumakovitch, Yongwu Rong and Valentina Harizanov, all of whom are at George Washington University.^[1]

This conference has become an important topological event in the Washington Metropolitan Area and regularly attracts well-known topologists from other areas of the US and from other countries.^[2] For example, Knots in Washington XVIII, held in May 2004, was the first conference fully devoted to the Khovanov homology, with Mikhail Khovanov giving a series of talks and leading experts Dror Bar-Natan, Lev Rozansky, Oleg Viro, and Ciprian Manolescu giving plenary talks.^[3] Knots in Washington XX was dedicated to the 60th birthday of Louis H. Kauffman.^[4] Other related conferences include Knots in Poland (1995, 2003),^[5] and Knots in Hellas in 1998, where Fields Medal winner Vaughan Jones spoke about his work on knot invariants.^[6]

Knots in Washington 50 will take place Dec 6th-8th, 2024 at the George Washington University with updated website: https://blogs.gwu.edu/ccas-knotsinwashington/

• Jozef Przytycki Archived 2009-03-01 at the Wayback Machine
• Knots in Washington web page

In knot theory, each link and knot can have an assigned knot thickness. Each realization of a link or knot has a thickness assigned to it. The thickness τ of a link allows us to introduce a scale with respect to which we can then define the ropelength of a link.

There exist several possible definitions of thickness that coincide for smooth enough curves.

Knotless knot
A knotless knot joining a fishing line (blue) to a fishing hook (grey) and a hair rig (orange)
Category	Hitch
Related	Snell knot
Typical use	Angling

The knotless knot is a hitch knot used to attach an eyed fishing hook to fishing line while leaving a length of line hanging below the hook.^[1] The extra length of line can then be used as the hair of a hair rig.^[2]

The knotless knot is usually tied to a leader, rather than directly to the main line. The far end of the leader may already be baited or looped in preparation for bait, because the knot will be tied using the near end as the working end.

The working end is passed through the hook eye from back to front (towards the hook point), wrapped tightly around itself and the hook shaft several times (down from the eye towards the bend of the hook), and finally passed once more through the eye back to front.^[3]

• Photographs showing how to tie the knotless knot
• Fishing Resources For Winter
• Illustrated instructions

In knot theory, a knot move or operation is a change or changes which preserve crossing number.^[1] Operations are used to investigate whether knots are equivalent, prime or reduced.

Knot moves or operations include the flype, Habiro move, Markov moves (I. conjugation and II. stabilization), pass move, Perko move, and Reidemeister moves (I. twist move, II. poke move, and III. slide move).^[1]

• Knot sum
• Murasugi sum
• Mutation (knot theory)

In physical knot theory, a knot energy is a functional on the space of all knot conformations. A conformation of a knot is a particular embedding of a circle into three-dimensional space. Depending on the needs of the energy function, the space of conformations is restricted to a sufficiently nicely behaved class. For example, one may consider only polygonal circles or C² functions. A property of the functional often requires that evolution of the knot under gradient descent does not change knot type.

Let ${\displaystyle X}$ be a subspace of ${\displaystyle C(\mathbb {S} ^{1},\mathbb {R} ^{3})}$ or of ${\displaystyle C(\mathbb {S} ^{1},\mathbb {S} ^{3})}$ with a topology (for example ${\displaystyle C^{1}}$ or some Sobolev-space with appropriate regularity) and let ${\displaystyle {\mathcal {K}}_{X}:=\{\gamma \in X:\gamma {\text{ is a topological embedding}}\}}$. A functional ${\displaystyle F:{\mathcal {K}}_{X}\rightarrow {\overline {\mathbb {R} }}}$ is called self-repulsive with respect topology on ${\displaystyle X}$ if ${\displaystyle F(\gamma _{n})\rightarrow +\infty }$ for all sequences ${\displaystyle (\gamma _{n})_{n\in \mathbb {N} }\subset {\mathcal {K}}_{X}}$ converging to an immersion with double point with respect to the topology on ${\displaystyle X}$.

A functional ${\displaystyle F:{\mathcal {K}}_{X}\rightarrow {\overline {\mathbb {R} }}}$ is a knot energy if and only if ${\displaystyle F}$ is a self-repulsive functional and bounded from below.

The most common type of knot energy comes from the intuition of the knot as electrically charged. Coulomb’s law states that two electric charges of the same sign will repel each other as the inverse square of the distance. Thus the knot will evolve under gradient descent according to the electric potential to an ideal configuration that minimizes the electrostatic energy. Naively defined, the integral for the energy will diverge and a regularization trick from physics, subtracting off a term from the energy, is necessary. In addition the knot could change knot type under evolution unless self-intersections are prevented.

An electrostatic energy of polygonal knots was studied by Fukuhara in 1987^[1] and shortly after a different, geometric energy was studied by Sakuma.^[2][3] In 1988, Jun O’Hara defined a knot energy based on electrostatic energy, Möbius energy.^[4] A fundamental property of the O’Hara energy function is that infinite energy barriers exist for passing the knot through itself. With some additional restrictions, O’Hara showed there were only finitely many knot types with energies less than a given bound. Later, Freedman, He, and Wang removed these restrictions.^[5]

Another type of knot energy arises from a more geometric idea. For example, the tangent point energies, first defined by Gonzalez and Maddocks.^[6] There, one double-integrate the inverse of the radius of the smallest circle being tangent at one point and passing through another point over the whole curve.^[7]

A similar kind of knot energies is given by the integral Menger curvature. There, one investigates the inverse of the radius the circle passing through three points of the knot and integrates this (three times) over the whole knot.

In knot theory, the Kinoshita–Terasaka knot is a particular prime knot with 11 crossings.^[1] It is named after Japanese mathematicians Shinichi Kinoshita and Hidetaka Terasaka, who wrote about it in 1957.^[2] The Kinoshita–Terasaka knot has a variety of interesting mathematical properties.^[3] It is related by mutation to the Conway knot,^[4] with which it shares a Jones polynomial. It has the same Alexander polynomial as the unknot.^[5]

• K11n42 at Knot Atlas

In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial.

It was developed in the late 1990s by Mikhail Khovanov.

To any link diagram ${\displaystyle D}$ representing a link ${\displaystyle L}$, we assign the Khovanov bracket ${\displaystyle \left[D\right]}$, a cochain complex of graded vector spaces. This is the analogue of the Kauffman bracket in the construction of the Jones polynomial. Next, we normalise ${\displaystyle \left[D\right]}$ by a series of degree shifts (in the graded vector spaces) and height shifts (in the cochain complex) to obtain a new cochain complex ${\displaystyle C(D)}$. The cohomology of this cochain complex turns out to be an invariant of ${\displaystyle L}$, and its graded Euler characteristic is the Jones polynomial of ${\displaystyle L}$.

This definition follows the formalism given in Dror Bar-Natan’s 2002 paper.

Let ${\displaystyle l}$ denote the degree shift operation on graded vector spaces—that is, the homogeneous component in dimension ${\displaystyle m}$ is shifted up to dimension ${\displaystyle m+l}$.

Similarly, let ${\displaystyle [s]}$ denote the height shift operation on cochain complexes—that is, the ${\displaystyle r}$th vector space or module in the complex is shifted along to the ${\displaystyle (r+s)}$th place, with all the differential maps being shifted accordingly.

Let ${\displaystyle V}$ be a graded vector space with one generator ${\displaystyle q}$ of degree 1, and one generator ${\displaystyle q^{-1}}$ of degree ${\displaystyle -1}$.

Now take an arbitrary diagram ${\displaystyle D}$ representing a link ${\displaystyle L}$. The axioms for the Khovanov bracket are as follows:

1. ${\displaystyle [\emptyset ]=(0\to \mathbb {Z} \to 0)}$, where ${\displaystyle \emptyset }$ denotes the empty link.
2. ${\displaystyle [{\text{O }}D]=V\otimes [D]}$, where O denotes an unlinked trivial component.
3. ${\displaystyle [D]=\mathbf {F} (0\to [D_{0}]\to [D_{1}]\{1\}\to 0)}$

In the third of these, ${\displaystyle \mathbf {F} }$ denotes the `flattening’ operation, where a single complex is formed from a double complex by taking direct sums along the diagonals. Also, ${\displaystyle D_{0}}$ denotes the `0-smoothing’ of a chosen crossing in ${\displaystyle D}$, and ${\displaystyle D_{1}}$ denotes the `1-smoothing’, analogously to the skein relation for the Kauffman bracket.

Next, we construct the `normalised’ complex ${\displaystyle \mathbf {C} (D)=[D][-n_{-}]\{n_{+}-2n_{-}\}}$, where ${\displaystyle n_{-}}$ denotes the number of left-handed crossings in the chosen diagram for ${\displaystyle D}$, and ${\displaystyle n_{+}}$ the number of right-handed crossings.

The Khovanov homology of ${\displaystyle L}$ is then defined as the cohomology ${\displaystyle \mathbf {H} (L)}$ of this complex ${\displaystyle \mathbf {C} (D)}$. It turns out that the Khovanov homology is indeed an invariant of ${\displaystyle L}$, and does not depend on the choice of diagram. The graded Euler characteristic of ${\displaystyle \mathbf {H} (L)}$ turns out to be the Jones polynomial of ${\displaystyle L}$. However, ${\displaystyle \mathbf {H} (L)}$ has been shown to contain more information about ${\displaystyle L}$ than the Jones polynomial, but the exact details are not yet fully understood.

In 2006 Dror Bar-Natan developed a computer program to calculate the Khovanov homology (or category) for any knot.^[1]

One of the most interesting aspects of Khovanov’s homology is that its exact sequences are formally similar to those arising in the Floer homology of 3-manifolds. Moreover, it has been used to produce another proof of a result first demonstrated using gauge theory and its cousins: Jacob Rasmussen’s new proof of a theorem of Peter Kronheimer and Tomasz Mrowka, formerly known as the Milnor conjecture (see below). There is a spectral sequence relating Khovanov homology with the knot Floer homology of Peter Ozsváth and Zoltán Szabó (Dowlin 2018).^[2] This spectral sequence settled an earlier conjecture on the relationship between the two theories (Dunfield et al. 2005). Another spectral sequence (Ozsváth-Szabó 2005) relates a variant of Khovanov homology with the Heegaard Floer homology of the branched double cover along a knot. A third (Bloom 2009) converges to a variant of the monopole Floer homology of the branched double cover. In 2010 Kronheimer and Mrowka ^[3] exhibited a spectral sequence abutting to their instanton knot Floer homology group and used this to show that Khovanov Homology (like the instanton knot Floer homology) detects the unknot.

Khovanov homology is related to the representation theory of the Lie algebra ${\displaystyle {\mathfrak {sl}}_{2}}$. Mikhail Khovanov and Lev Rozansky have since defined homology theories associated to ${\displaystyle {\mathfrak {sl}}_{n}}$ for all ${\displaystyle n}$. In 2003, Catharina Stroppel extended Khovanov homology to an invariant of tangles (a categorified version of Reshetikhin-Turaev invariants) which also generalizes to ${\displaystyle {\mathfrak {sl}}_{n}}$ for all ${\displaystyle n}$.
Paul Seidel and Ivan Smith have constructed a singly graded knot homology theory using Lagrangian intersection Floer homology, which they conjecture to be isomorphic to a singly graded version of Khovanov homology. Ciprian Manolescu has since simplified their construction and shown how to recover the Jones polynomial from the cochain complex underlying his version of the Seidel–Smith invariant.

At International Congress of Mathematicians in 2006 Mikhail Khovanov provided the following explanation for the relation to knot polynomials from the view point of Khovanov homology. The skein relation for three links ${\displaystyle L_{1},L_{2}}$ and ${\displaystyle L_{3}}$ is described as

${\displaystyle \lambda P(L_{1})-\lambda ^{-1}P(L_{2})=(q-q^{-1})P(L_{3}).}$

Substituting ${\displaystyle \lambda =q^{n},n\leq 0}$ leads to a link polynomial invariant ${\displaystyle P_{n}(L)\in \mathbb {Z} [q,q^{-1}]}$, normalized so that

${\displaystyle {\begin{aligned}P_{n}(unknot)&=q^{n-1}+q^{n-3}+\cdots +q^{1-n}&&n>0\\P_{0}(unknot)&=1\end{aligned}}}$

For ${\displaystyle n>1}$ the polynomial ${\displaystyle P_{n}(L)}$ can be interpreted via the representation theory of quantum group ${\displaystyle U_{q}(sl(n))}$ and ${\displaystyle P_{0}(L)}$ via that of the quantum Lie superalgebra ${\displaystyle U_{q}(gl(1|1))}$.

• The Alexander polynomial ${\displaystyle P_{0}(L)}$ is the Euler characteristic of a bigraded knot homology theory.
• ${\displaystyle P_{1}(L)=1}$ is trivial.
• The Jones polynomial ${\displaystyle P_{2}(L)}$ is the Euler characteristic of a bigraded link homology theory.
• The entire HOMFLY-PT polynomial is the Euler characteristic of a triply graded link homology theory.

The first application of Khovanov homology was provided by Jacob Rasmussen, who defined the s-invariant using Khovanov homology. This integer valued invariant of a knot gives a bound on the slice genus, and is sufficient to prove the Milnor conjecture.

In 2010, Kronheimer and Mrowka proved that the Khovanov homology detects the unknot. The categorified theory has more information than the non-categorified theory. Although the Khovanov homology detects the unknot, it is not yet known if the Jones polynomial does.

• Bar-Natan, Dror (2002), “On Khovanov’s categorification of the Jones polynomial”, Algebraic & Geometric Topology, 2: 337–370, arXiv:math.QA/0201043, Bibcode:2002math……1043B, doi:10.2140/agt.2002.2.337, MR 1917056, S2CID 11754112.
• Bloom, Jonathan M. (2011), “A link surgery spectral sequence in monopole Floer homology”, Advances in Mathematics, 226 (4): 3216–3281, arXiv:0909.0816, doi:10.1016/j.aim.2010.10.014, MR 2764887, S2CID 11791207.
• Dunfield, Nathan M.; Gukov, Sergei; Rasmussen, Jacob (2006), “The superpolynomial for knot homologies”, Experimental Mathematics, 15 (2): 129–159, arXiv:math.GT/0505662, doi:10.1080/10586458.2006.10128956, MR 2253002, S2CID 3060662.
• Khovanov, Mikhail (2000), “A categorification of the Jones polynomial”, Duke Mathematical Journal, 101 (3): 359–426, arXiv:math.QA/9908171, doi:10.1215/S0012-7094-00-10131-7, MR 1740682, S2CID 119585149.
• Khovanov, Mikhail (2006), “Link homology and categorification”, International Congress of Mathematicians. Vol. II, Zürich: European Mathematical Society, pp. 989–999, arXiv:math.GT/0605339, MR 2275632.
• Ozsváth, Peter; Szabó, Zoltán (2005), “On the Heegaard Floer homology of branched double-covers”, Advances in Mathematics, 194 (1): 1–33, arXiv:math.GT/0309170, doi:10.1016/j.aim.2004.05.008, MR 2141852, S2CID 17245314.
• Stroppel, Catharina (2005), “Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors”, Duke Mathematical Journal, 126 (3): 547–596, CiteSeerX 10.1.1.586.3553, doi:10.1215/S0012-7094-04-12634-X, MR 2120117.

• Hand-written slides of M. Khovanov’s talk
• “Khovanov Homology“, The Knot Atlas.

Karash double loop
Category	Loop
Efficiency	49-72% (12.5 mm static)[1]
Typical use	climbing, rope rescue

Karash double loop is a common name for a knot forming two loops. This knot has been a known variant of the Bowline on a bight per the International Guild of Knot Tyers, referred to as bowline twist or twisted collar bowline on a bight.^[2] The knot is also referred to as nœud de fusion in French references and sometimes called Fusion knot in English.

The name Karash double loop was introduced by Mike Karash, who re-invented the knot to create makeshift harnesses for rescue operations and popularized it among rescue workers.^[3][4]

The knot is used for vertical caving using the single rope technique,^[5] particularly by French cavers. It is advertised by the French Federation of Speleology as a safe alternative to the bowline on a bight. Compared to the traditional double loop variant of the figure of eight, its loops remain open under load allowing to clip and unclip carabiners in the loops more easily.

The knot is also popular to create a makeshift harness in rescue operations. The two loops are used as leg loops to sit in. The knot can be further improved by adding two bowlines around a person’s body to create a three-point harness.^[6][7]

The traditional method of tying this knot starts with a reverse loop (like the Eskimo bowline) then wraps around the standing end (AKA the “tree”), you then finish the knot the same way as the BOAB – Bowline on a bight.

An alternative way as advertised by the French Federation of Speleology and Mike Karash is starting with a figure of eight on a bight, then pulls two strands of the rope back through the bight.

• 
  Start with a bight of rope
• 
  Tie a figure of eight on the bight
• 
• 
  Pull the loop of the figure of eight over the whole knot
• 
• 
• 
  Take the two strands of rope together
• 
  Pull back the two strands of rope closest to the loop inside the knot
• 
  These two form the final two loops of the knot
• 
  Tighten and dress the knot

• Official website

Jury mast knot
Three variations of the Jury mast knot
Names	Jury mast knot, Masthead knot, Pitcher Knot, Jury masthead
Category	Loop
Related	Tom fool’s knot, Handcuff knot, Bottle sling
Typical use	Jury rigging a mast, carrying pitchers
ABoK	#1167, #1168, #1169, #2563

The jury mast knot (or masthead knot) is traditionally presented as to be used for jury rigging a temporary mast on a sailboat or ship after the original one has been lost; some authors claim a use for derrick poles—but there is no good evidence for actual use. The knot is placed at the top of a new mast with the mast projecting through the center of the knot. The loops of the knot are then used as anchor points for makeshift stays and shrouds. Usually small blocks of wood are affixed to, or a groove cut in, the new mast to prevent the knot from sliding downwards.^[1]

Due to a lack of hard historical evidence there is uncertainty whether this supposed rigging knot was ever commonly used for rigging jury masts.^[2]

There are three closely related variations of this knot. They differ based on the type of crossing, overhand or underhand, of the three initial loops and then whether the edges of each loop is positioned over or under the previous one. Although these knots are tied in the bight, for the purposes of description the left side will be considered the standing part. If all the crossings and overlays are reversed, or the right side is taken as the standing part, a mirror image of the knot will result.

#1167

This variation might grip the mast best (per Ashley) in the absence of other means to prevent the knot from sliding downwards. The pattern of this variation, from the left, is: under-under/over-over.

#1168

This variation is the simplest of the three in structure (though not in tying) and lends itself to being reinforced by tying a reef knot with the two free ends. The pattern of this variation, from the left, is: under-over-under/over-over.

#1169

The pattern of this variation, from the left, is: under-under-under/under-under.

• Description of tying a Masthead knot

Improved clinch knot
Names	Improved clinch knot, fisherman’s knot, salmon knot
Category	Hitch
Efficiency	98%
Origin	Unknown
Typical use	fishing, angling, trapping
ABoK	#313

The improved clinch knot, also known as the fisherman’s knot^[1] or the salmon knot,^[2] is a knot that is used for securing a fishing line to the fishing lure, but can also affix fishing line to a swivel, clip, or artificial fly. This is a common knot used by anglers because of its simple tie and strong hold. The more tension is applied, the tighter the knot becomes, increasing the strength of the connection. It can be used with many kinds of line including mono-filament, fluorocarbon, and braided fishing line. The difference from the basic clinch knot is that the working end is passed through the loop that is created in the second-last step.^[3]

• List of knots

• Grog. “Improved clinch knot”. Animated Knots. Retrieved 25 April 2013.