Invertible knot

Last updated

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 with its orientation reversed. A non-invertible knot is any knot which does not have this property. The invertibility of a knot is a knot invariant. An invertible link is the link equivalent of an invertible knot.

Contents

There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, reversible, positively amphichiral noninvertible, negatively amphichiral noninvertible, and fully amphichiral invertible. [1]

Background

Number of invertible and non-invertible knots for each crossing number
Number of crossings345678910111213141516 OEIS sequence
Non-invertible knots00000123318711446919381182265811309875 A052402
Invertible knots1123720471323651032306988542671278830 A052403

It has long been known that most of the simple knots, such as the trefoil knot and the figure-eight knot are invertible. In 1962 Ralph Fox conjectured that some knots were non-invertible, but it was not proved that non-invertible knots exist until Hale Trotter discovered an infinite family of pretzel knots that were non-invertible in 1963. [2] It is now known almost all knots are non-invertible. [3]

Invertible knots

The simplest non-trivial invertible knot, the trefoil knot. Rotating the knot 180 degrees in 3-space about an axis in the plane of the diagram produces the same knot diagram, but with the arrow's direction reversed. Knot-trefoil-dir-128.png
The simplest non-trivial invertible knot, the trefoil knot. Rotating the knot 180 degrees in 3-space about an axis in the plane of the diagram produces the same knot diagram, but with the arrow's direction reversed.

All knots with crossing number of 7 or less are known to be invertible. No general method is known that can distinguish if a given knot is invertible. [4] The problem can be translated into algebraic terms, [5] but unfortunately there is no known algorithm to solve this algebraic problem.

If a knot is invertible and amphichiral, it is fully amphichiral. The simplest knot with this property is the figure eight knot. A chiral knot that is invertible is classified as a reversible knot. [6]

Strongly invertible knots

A more abstract way to define an invertible knot is to say there is an orientation-preserving homeomorphism of the 3-sphere which takes the knot to itself but reverses the orientation along the knot. By imposing the stronger condition that the homeomorphism also be an involution, i.e. have period 2 in the homeomorphism group of the 3-sphere, we arrive at the definition of a strongly invertible knot. All knots with tunnel number one, such as the trefoil knot and figure-eight knot, are strongly invertible. [7]

Non-invertible knots

The non-invertible knot 817, the simplest of the non-invertible knots. 8 17 Knot.svg
The non-invertible knot 817, the simplest of the non-invertible knots.

The simplest example of a non-invertible knot is the knot 817 (Alexander-Briggs notation) or .2.2 (Conway notation). The pretzel knot 7, 5, 3 is non-invertible, as are all pretzel knots of the form (2p + 1), (2q + 1), (2r + 1), where p, q, and r are distinct integers, which is the infinite family proven to be non-invertible by Trotter. [2]

See also

Related Research Articles

<span class="mw-page-title-main">Algebraic topology</span> Branch of mathematics

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

<span class="mw-page-title-main">Figure-eight knot (mathematics)</span> Unique knot with a crossing number of four

In knot theory, a figure-eight knot 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.

<span class="mw-page-title-main">Knot theory</span> Study of mathematical knots

In the mathematical field of 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, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.

<span class="mw-page-title-main">Trefoil knot</span> 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 together 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.

In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.

<span class="mw-page-title-main">Borromean rings</span> Three linked but pairwise separated rings

In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed. Most commonly, these rings are drawn as three circles in the plane, in the pattern of a Venn diagram, alternatingly crossing over and under each other at the points where they cross. Other triples of curves are said to form the Borromean rings as long as they are topologically equivalent to the curves depicted in this drawing.

<span class="mw-page-title-main">Lens space</span> 3-manifold that is a quotient of S³ by ℤ/p actions: (z,w) ↦ (exp(2πi/p)z, exp(2πiq/p)w)

A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions.

In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one.

<span class="mw-page-title-main">Chirality (mathematics)</span> Property of an object that is not congruent to its mirror image

In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be achiral.

<span class="mw-page-title-main">Pretzel link</span> Link formed from a finite number of twisted sections

In the mathematical theory of knots, a pretzel link is a special kind of link. It consists of a finite number tangles made of two intertwined circular helices. The tangles are connected cyclicly, the first component of the first tangle is connected to the second component of the second tangle, etc., with the first component of the last tangle connected to the second component of the first. A pretzel link which is also a knot is a pretzel knot.

<span class="mw-page-title-main">Fibered knot</span> Mathematical knot

In knot theory, a branch of mathematics, a knot or link in the 3-dimensional sphere is called fibered or fibred if there is a 1-parameter family of Seifert surfaces for , where the parameter runs through the points of the unit circle , such that if is not equal to then the intersection of and is exactly .

<span class="mw-page-title-main">Hopf link</span> Simplest nontrivial knot link

In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf.

The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe. All of the Tait conjectures have been solved, the most recent being the Flyping conjecture.

In the mathematical field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image. An oriented knot that is equivalent to its mirror image is an amphicheiral knot, also called an achiral knot. The chirality of a knot is a knot invariant. A knot's chirality can be further classified depending on whether or not it is invertible.

<span class="mw-page-title-main">Ribbon knot</span>

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 kind of singularity can be formed by cutting a slit in the disk and passing another part of the disk through the slit. More precisely, this type of singularity is a closed arc consisting of intersection points of the disk with itself, such that the preimage of this arc consists of two arcs in the disc, one completely in the interior of the disk and the other having its two endpoints on the disk boundary.

<span class="mw-page-title-main">Ropelength</span> Knot invariant

In physical knot theory, each realization of a link or knot has an associated ropelength. Intuitively this is the minimal length of an ideally flexible rope that is needed to tie a given link, or knot. Knots and links that minimize ropelength are called ideal knots and ideal links respectively.

<span class="mw-page-title-main">Hale Trotter</span> Canadian-American mathematician (1931–2022)

Hale Freeman Trotter was a Canadian-American mathematician, known for the Lie–Trotter product formula, the Steinhaus–Johnson–Trotter algorithm, and the Lang–Trotter conjecture. He was born in Kingston, Ontario. He died in Princeton, New Jersey on January 17, 2022.

<span class="mw-page-title-main">Dan Burghelea</span> Romanian-American mathematician

Dan Burghelea is a Romanian-American mathematician, academic, and researcher. He is an Emeritus Professor of Mathematics at Ohio State University.

References

  1. Hoste, Jim; Thistlethwaite, Morwen; Weeks, Jeff (1998), "The first 1,701,936 knots" (PDF), The Mathematical Intelligencer, 20 (4): 33–48, doi:10.1007/BF03025227, MR   1646740, archived from the original (PDF) on 2013-12-15.
  2. 1 2 Trotter, H. F. (1963), "Non-invertible knots exist", Topology, 2: 275–280, doi: 10.1016/0040-9383(63)90011-9 , MR   0158395 .
  3. Murasugi, Kunio (2007), Knot Theory and Its Applications, Springer, p. 45, ISBN   9780817647186 .
  4. Weisstein, Eric W. "Invertible Knot". MathWorld . Accessed: May 5, 2013.
  5. Kuperberg, Greg (1996), "Detecting knot invertibility", Journal of Knot Theory and its Ramifications, 5 (2): 173–181, arXiv: q-alg/9712048 , doi:10.1142/S021821659600014X, MR   1395778 .
  6. Clark, W. Edwin; Elhamdadi, Mohamed; Saito, Masahico; Yeatman, Timothy (2013), Quandle colorings of knots and applications, arXiv: 1312.3307 , Bibcode:2013arXiv1312.3307C .
  7. Morimoto, Kanji (1995), "There are knots whose tunnel numbers go down under connected sum", Proceedings of the American Mathematical Society, 123 (11): 3527–3532, doi: 10.1090/S0002-9939-1995-1317043-4 , JSTOR   2161103, MR   1317043 . See in particular Lemma 5.