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Pretzel bread in the shape of a 7 4 pretzel knot. In mathematics, a knot is an embedding of the circle (S 1) into three-dimensional Euclidean space, R 3 (also known as E 3). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of R 3 which takes one knot to the other.
In 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 (or "unknot").
In Knot theory 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 perturbation in the choice 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 [3]
3 1 knot/Trefoil knot - (2,3)-torus knot, the two loose ends of a common overhand knot joined together 4 1 knot/ Figure-eight knot (mathematics) - a prime knot with a crossing number four 5 1 knot/ Cinquefoil knot , (5,2)-torus knot, Solomon's seal knot, pentafoil knot - a prime knot with crossing number five which can be arranged as a {5/2 ...
Figure-eight knot (mathematics) the only 4-crossing knot; Granny knot (mathematics) and Square knot (mathematics) are a connected sum of two Trefoil knots; Perko pair, two entries in a knot table that were later shown to be identical. Stevedore knot (mathematics), a prime knot with crossing number 6; Three-twist knot is the twist knot with ...
The first work of knot theory to include the Borromean rings was a catalog of knots and links compiled in 1876 by Peter Tait. [3] In recreational mathematics, the Borromean rings were popularized by Martin Gardner, who featured Seifert surfaces for the Borromean rings in his September 1961 "Mathematical Games" column in Scientific American. [19]
A knot diagram with crossings labelled for a Dowker sequence. In the mathematical field of knot theory, the Dowker–Thistlethwaite (DT) notation or code, for a knot is a sequence of even integers. The notation is named after Clifford Hugh Dowker and Morwen Thistlethwaite, who refined a notation originally due to Peter Guthrie Tait. [1]
In knot theory, a chord diagram can be used to describe the sequence of crossings along the planar projection of a knot, with each point at which a crossing occurs paired with the point that crosses it. To fully describe the knot, the diagram should be annotated with an extra bit of information for each pair, indicating which point crosses over ...