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In knot theory
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 ...
, Conway notation, invented by John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branc ...
, is a way of describing knot
A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...
s that makes many of their properties clear. It composes a knot using certain operations on tangle
Tangle may refer to: Science, Technology, Engineering & Mathematics
*''The Tangle'' is the name of the ledger, a directed acyclic graph, used for the cryptocurrency IOTA
*Tangle (mathematics), a topological object
Natural sciences & medicine ...
s to construct it.
Basic concepts
Tangles
In Conway notation, thetangle
Tangle may refer to: Science, Technology, Engineering & Mathematics
*''The Tangle'' is the name of the ledger, a directed acyclic graph, used for the cryptocurrency IOTA
*Tangle (mathematics), a topological object
Natural sciences & medicine ...
s are generally algebraic 2-tangles. This means their tangle diagrams consist of 2 arcs and 4 points on the edge of the diagram; furthermore, they are built up from rational tangle
In mathematics, a tangle is generally one of two related concepts:
* In John Conway's definition, an ''n''-tangle is a proper embedding of the disjoint union of ''n'' arcs into a 3-ball; the embedding must send the endpoints of the arcs to 2''n' ...
s using the Conway operations.
he following seems to be attempting to describe only integer or 1/n rational tanglesTangles consisting only of positive crossings are denoted by the number of crossings, or if there are only negative crossings it is denoted by a negative number. If the arcs are not crossed, or can be transformed into an uncrossed position with the Reidemeister move
Kurt Werner Friedrich Reidemeister (13 October 1893 – 8 July 1971) was a mathematician born in Braunschweig (Brunswick), Germany.
Life
He was a brother of Marie Neurath.
Beginning in 1912, he studied in Freiburg, Munich, Marburg, and Götting ...
s, it is called the 0 or ∞ tangle, depending on the orientation of the tangle.
Operations on tangles
If a tangle, ''a'', is reflected on the NW-SE line, it is denoted by ''−a''. (Note that this is different from a tangle with a negative number of crossings.)Tangles have threebinary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
s, ''sum'', ''product'', and ''ramification'', however all can be explained using tangle addition and negation. The tangle product, ''a b'', is equivalent to ''−a+b''. and ramification or ''a,b'', is equivalent to ''−a+−b''.
Advanced concepts
Rational tangles are equivalentif and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
their fractions are equal. An accessible proof of this fact is given in (Kauffman and Lambropoulou 2004). A number before an asterisk, ''*'', denotes the polyhedron number; multiple asterisks indicate that multiple polyhedra of that number exist.
See also
*Conway knot
In mathematics, in particular in knot theory, the Conway knot (or Conway's knot) is a particular knot with 11 crossings, named after John Horton Conway.
It is related by mutation to the Kinoshita–Terasaka knot, with which it shares the sam ...
*Dowker notation Dowker is a surname. Notable people with the surname include:
* Clifford Hugh Dowker (1912–1982), Canadian mathematician
*Fay Dowker (born 1965), British physicist
* Felicity Dowker, Australian fantasy writer
* Hasted Dowker (1900–1986), Canadi ...
*Alexander–Briggs notation
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 ...
*Gauss notation
Gauss notation (also known as a Gauss code or Gauss word) is a notation for mathematical knots. It is created by enumerating and classifying the crossings of an embedding of the knot in a plane. It is named for the mathematician Carl Friedrich Ga ...
References
Further reading
* * {{Knot theory, state=collapsed Knot theory John Horton Conway