Knots (de Gruyter Studies in Mathematics)

Higher-dimensional knots are n -dimensional spheres in m -dimensional Euclidean space. Archaeologists have discovered that knot tying dates back to prehistoric times. Besides their uses such as recording information and tying objects together, knots have interested humans for their aesthetics and spiritual symbolism. Knots appear in various forms of Chinese artwork dating from several centuries BC see Chinese knotting. The endless knot appears in Tibetan Buddhism , while the Borromean rings have made repeated appearances in different cultures, often representing strength in unity. The Celtic monks who created the Book of Kells lavished entire pages with intricate Celtic knotwork.

Mathematical studies of knots began in the 19th century with Carl Friedrich Gauss , who defined the linking integral Silver In the s, Lord Kelvin 's theory that atoms were knots in the aether led to Peter Guthrie Tait 's creation of the first knot tables for complete classification. Tait, in , published a table of knots with up to ten crossings, and what came to be known as the Tait conjectures. This record motivated the early knot theorists, but knot theory eventually became part of the emerging subject of topology.

These topologists in the early part of the 20th century— Max Dehn , J. Alexander , and others—studied knots from the point of view of the knot group and invariants from homology theory such as the Alexander polynomial. This would be the main approach to knot theory until a series of breakthroughs transformed the subject.

In the late s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem.

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Many knots were shown to be hyperbolic knots , enabling the use of geometry in defining new, powerful knot invariants. The discovery of the Jones polynomial by Vaughan Jones in Sossinsky , pp.

A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as quantum groups and Floer homology. In the last several decades of the 20th century, scientists became interested in studying physical knots in order to understand knotting phenomena in DNA and other polymers.

Knot theory can be used to determine if a molecule is chiral has a "handedness" or not Simon Tangles , strings with both ends fixed in place, have been effectively used in studying the action of topoisomerase on DNA Flapan Knot theory may be crucial in the construction of quantum computers, through the model of topological quantum computation Collins A knot is created by beginning with a one- dimensional line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop Adams Sossinsky Topologists consider knots and other entanglements such as links and braids to be equivalent if the knot can be pushed about smoothly, without intersecting itself, to coincide with another knot.

The idea of knot equivalence is to give a precise definition of when two knots should be considered the same even when positioned quite differently in space. Two knots K 1 and K 2 are equivalent if there exists a continuous mapping H: Such a function H is known as an ambient isotopy. These two notions of knot equivalence agree exactly about which knots are equivalent: Two knots that are equivalent under the orientation-preserving homeomorphism definition are also equivalent under the ambient isotopy definition, because any orientation-preserving homeomorphisms of R 3 to itself is the final stage of an ambient isotopy starting from the identity.

The basic problem of knot theory, the recognition problem , is determining the equivalence of two knots. Algorithms exist to solve this problem, with the first given by Wolfgang Haken in the late s Hass Nonetheless, these algorithms can be extremely time-consuming, and a major issue in the theory is to understand how hard this problem really is Hass The special case of recognizing the unknot , called the unknotting problem , is of particular interest Hoste A useful way to visualise and manipulate knots is to project the knot onto a plane—think of the knot casting a shadow on the wall.

A small change in the direction of projection will ensure that it is one-to-one except at the double points, called crossings , where the "shadow" of the knot crosses itself once transversely Rolfsen At each crossing, to be able to recreate the original knot, the over-strand must be distinguished from the under-strand. This is often done by creating a break in the strand going underneath. The resulting diagram is an immersed plane curve with the additional data of which strand is over and which is under at each crossing.

These diagrams are called knot diagrams when they represent a knot and link diagrams when they represent a link. Analogously, knotted surfaces in 4-space can be related to immersed surfaces in 3-space. A reduced diagram is a knot diagram in which there are no reducible crossings also nugatory or removable crossings , or in which all of the reducible crossings have been removed. In , working with this diagrammatic form of knots, J.

Alexander and Garland Baird Briggs, and independently Kurt Reidemeister , demonstrated that two knot diagrams belonging to the same knot can be related by a sequence of three kinds of moves on the diagram, shown below. These operations, now called the Reidemeister moves , are:.

The proof that diagrams of equivalent knots are connected by Reidemeister moves relies on an analysis of what happens under the planar projection of the movement taking one knot to another. The movement can be arranged so that almost all of the time the projection will be a knot diagram, except at finitely many times when an "event" or "catastrophe" occurs, such as when more than two strands cross at a point or multiple strands become tangent at a point.

A close inspection will show that complicated events can be eliminated, leaving only the simplest events: These are precisely the Reidemeister moves Sossinsky , ch. A knot invariant is a "quantity" that is the same for equivalent knots Adams Lickorish Rolfsen For example, if the invariant is computed from a knot diagram, it should give the same value for two knot diagrams representing equivalent knots. An invariant may take the same value on two different knots, so by itself may be incapable of distinguishing all knots. An elementary invariant is tricolorability.

In the late 20th century, invariants such as "quantum" knot polynomials, Vassiliev invariants and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory. A knot polynomial is a knot invariant that is a polynomial.

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Well-known examples include the Jones and Alexander polynomials. A variant of the Alexander polynomial, the Alexander—Conway polynomial , is a polynomial in the variable z with integer coefficients Lickorish The Alexander—Conway polynomial is actually defined in terms of links , which consist of one or more knots entangled with each other. The concepts explained above for knots, e.

Consider an oriented link diagram, i. The second rule is what is often referred to as a skein relation. To check that these rules give an invariant of an oriented link, one should determine that the polynomial does not change under the three Reidemeister moves. Many important knot polynomials can be defined in this way. The following is an example of a typical computation using a skein relation.

It computes the Alexander—Conway polynomial of the trefoil knot. The yellow patches indicate where the relation is applied. Applying the relation to the Hopf link where indicated,. The unlink takes a bit of sneakiness:. Since the Alexander—Conway polynomial is a knot invariant, this shows that the trefoil is not equivalent to the unknot.

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In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.