Knot theory

upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.

For example, a higher-dimensional knot is an n-dimensional sphere embedded in (n+2)-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.

Mathematical studies of knots began in the 19th century with Carl Friedrich Gauss, who defined the linking integral (Silver 2006).

A plethora of knot invariants have been invented since then, utilizing sophisticated tools such as quantum groups and Floer homology.

Tangles, strings with both ends fixed in place, have been effectively used in studying the action of topoisomerase on DNA (Flapan 2000).

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 2004) (Sossinsky 2002).

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 1998).

In February 2021 Marc Lackenby announced a new unknot recognition algorithm that runs in quasi-polynomial time.

The resulting diagram is an immersed plane curve with the additional data of which strand is over and which is under at each crossing.

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: (1) a "kink" forming or being straightened out; (2) two strands becoming tangent at a point and passing through; and (3) three strands crossing at a point.

The concepts explained above for knots, e.g. diagrams and Reidemeister moves, also hold for links.

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.

The Jones polynomial can in fact distinguish between the left- and right-handed trefoil knots (Lickorish 1997).

The fundamental parallelogram (which is indicated in the picture), tiles both vertically and horizontally and shows how to extend the pattern of spheres infinitely.

Other hyperbolic invariants include the shape of the fundamental parallelogram, length of shortest geodesic, and volume.

Analogies for the plane would be lifting a string up off the surface, or removing a dot from inside a circle.

In fact, in four dimensions, any non-intersecting closed loop of one-dimensional string is equivalent to an unknot.

The mathematical technique called "general position" implies that for a given n-sphere in m-dimensional Euclidean space, if m is large enough (depending on n), the sphere should be unknotted.

This is done by combining basic units called soft contacts using five operations (Parallel, Series, Cross, Concerted, and Sub).

[8][9] The approach is applicable to open chains as well and can also be extended to include the so-called hard contacts.

This famous error would propagate when Dale Rolfsen added a knot table in his influential text, based on Conway's work.

Conway's 1970 paper on knot theory also contains a typographical duplication on its non-alternating 11-crossing knots page and omits 4 examples — 2 previously listed in D. Lombardero's 1968 Princeton senior thesis and 2 more subsequently discovered by Alain Caudron.

[see Perko (1982), Primality of certain knots, Topology Proceedings] Less famous is the duplicate in his 10 crossing link table: 2.-2.-20.20 is the mirror of 8*-20:-20.

Alexander–Briggs names in the range 10162 to 10166 are ambiguous, due to the discovery of the Perko pair in Charles Newton Little's original and subsequent knot tables, and differences in approach to correcting this error in knot tables and other publications created after this point.

Any link admits such a description, and it is clear this is a very compact notation even for very large crossing number.

Gauss code, similar to the Dowker–Thistlethwaite notation, represents a knot with a sequence of integers.

Lickorish is a rigorous introduction for graduate students, covering a nice mix of classical and modern topics.