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The knot (/ n ɒ t /) is a unit of ... would therefore be useless on such a chart. Since the length of a nautical mile, for practical purposes, is equivalent to about ...
A knot invariant is a "quantity" that is the same for equivalent knots (Lickorish 1997) (Rolfsen 1976). For example, if the invariant is computed from a knot diagram, it should give the same value for two knot diagrams representing equivalent knots.
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 ...
In knot theory, prime knots are those knots that are indecomposable under the operation of knot sum. The prime knots with ten or fewer crossings are listed here for quick comparison of their properties and varied naming schemes.
The (p,−q) torus knot is the obverse (mirror image) of the (p,q) torus knot. [5] The (−p,−q) torus knot is equivalent to the (p,q) torus knot except for the reversed orientation. The (3, 4) torus knot on the unwrapped torus surface, and its braid word. Any (p,q)-torus knot can be made from a closed braid with p strands.
7 1 knot, septafoil knot, (7,2)-torus knot - a prime knot with crossing number seven, which can be arranged as a {7/2} star polygon ; 7 4 knot, "endless knot" 8 18 knot, "carrick mat" 10 161 /10 162, known as the Perko pair; this was a single knot listed twice in Dale Rolfsen's knot table; the duplication was discovered by Kenneth Perko
An oriented knot that is equivalent to its mirror image is an amphicheiral knot, also called an achiral knot. The chirality of a knot is a knot invariant . A knot's chirality can be further classified depending on whether or not it is invertible .
Knots K 1 and K 2 are considered equivalent when there is an ambient isotopy which moves K 1 to K 2. This is the appropriate definition in the topological category. Similar language is used for the equivalent concept in contexts where one has a stronger notion of equivalence. For example, a path between two smooth embeddings is a smooth isotopy.