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in the opposite direction from before. The list begins with 0 if you start with two vertical arcs. The diagram with two horizontal arcs is then (0), but we assign (0, 0) to the diagram with vertical arcs. A convention is needed to describe a "positive" or "negative" twist. Often, "rational tangle" refers to a list of numbers representing a simple diagram as described.
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We can describe such a diagram by considering the numbers given by consecutive twists around the same set of endpoints, e.g. (2, 1, -3) means start with two horizontal arcs, then 2 twists using NE/SE endpoints, then 1 twist using SW/SE endpoints, and then 3 twists using NE/SE endpoints but twisting
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An arbitrary tangle diagram of a rational tangle may look very complicated, but there is always a diagram of a particular simple form: start with a tangle diagram consisting of two horizontal (vertical) arcs; add a "twist", i.e. a single crossing by switching the NE and SE endpoints (SW and SE
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is a 2-tangle that is homeomorphic to the trivial 2-tangle by a map of pairs consisting of the 3-ball and two arcs. The four endpoints of the arcs on the boundary circle of a tangle diagram are usually referred as NE, NW, SW, SE, with the symbols referring to the compass directions.
187:– the difference from the previous definition is that it includes circles as well as arcs, and partitions the boundary into two (isomorphic) pieces, which is algebraically more convenient – it allows one to add tangles by stacking them, for instance.
507:. Conway proved that the fraction is well-defined and completely determines the rational tangle up to tangle equivalence. An accessible proof of this fact is given in:. Conway also defined a fraction of an arbitrary tangle by using the
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endpoints); continue by adding more twists using either the NE and SE endpoints or the SW and SE endpoints. One can suppose each twist does not change the diagram inside a disc containing previously created crossings.
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There is an "arithmetic" of tangles with addition, multiplication, and reciprocal operations. An algebraic tangle is obtained from the addition and multiplication of rational tangles.
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One motivation for Conway's study of tangles was to provide a notation for knots more systematic than the traditional enumeration found in tables.
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of a rational tangle is defined as the link obtained by joining the "north" endpoints together and the "south" endpoints also together. The
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Without loss of generality, consider the marked points on the 3-ball boundary to lie on a great circle. The tangle can be arranged to be in
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Ernst, C.; Sumners, D. W. (November 1990). "A calculus for rational tangles: applications to DNA recombination".
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with respect to the projection onto the flat disc bounded by the great circle. The projection then gives us a
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Tangles often show up as tangle diagrams in knot or link diagrams and can be used as building blocks for
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B 52 (1991) 153–190, who used it to describe separation in graphs. This usage has been extended to
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The balance of this article discusses Conway's sense of tangles; for the link theory sense, see
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except, instead of closed loops, strings whose ends are nailed down are used. See also
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of one tangle to the other keeping the boundary of the 3-ball fixed.
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is defined similarly by grouping the "east" and "west" endpoints.
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is then defined as the number given by the continued fraction
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are defined to be such closures of rational tangles.
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