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Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram, in which any knot can be drawn in many different ways.
A polygonal knot is a knot whose image in R 3 is the union of a finite set of line segments. [6] A tame knot is any knot equivalent to a polygonal knot. [6] [Note 2] Knots which are not tame are called wild, [7] and can have pathological behavior. [7] In knot theory and 3-manifold theory, often the adjective "tame" is omitted. Smooth knots, for ...
A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory . Implicit in this definition is that there is a trivial reference link, usually called the unlink , but the word is also sometimes used in context where there is no notion of a trivial link.
Any reduced diagram of an alternating link has the fewest possible crossings. Any two reduced diagrams of the same alternating knot have the same writhe. Given any two reduced alternating diagrams D 1 and D 2 of an oriented, prime alternating link: D 1 may be transformed to D 2 by means of a sequence of certain simple moves called flypes. Also ...
Link diagrams must be considered because a single skein change can alter a diagram from representing a knot to one representing a link and vice versa. Depending on the knot polynomial in question, the links (or tangles) appearing in a skein relation may be oriented or unoriented. The three diagrams are labelled as follows.
Algebraic link diagram for the Borromean rings. The vertical dotted black midline is a Conway sphere separating the diagram into 2-tangles. In knot theory, the Borromean rings are a simple example of a Brunnian link, a link that cannot be separated but that falls apart into separate unknotted loops as soon as any one of its components is ...
Figure-eight knot of practical knot-tying, with ends joined. In knot theory, a figure-eight knot (also called Listing's knot [1]) 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.
In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram. Kurt Reidemeister () and, independently, James Waddell Alexander and Garland Baird Briggs (), demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves.