In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

knot theory In topology, knot theory is the study of knot (mathematics), 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 und ...

, the Dowker–Thistlethwaite (DT) notation or code, for a

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, ...

is a sequence of even

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s. The notation is named after Clifford Hugh Dowker and Morwen Thistlethwaite, who refined a notation originally due to

Peter Guthrie Tait Peter Guthrie Tait (28 April 18314 July 1901) was a Scottish Mathematical physics, mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook ''Treatise on Natural Philosophy'', which he ...

Definition

To generate the Dowker–Thistlethwaite notation, traverse the knot using an arbitrary starting point and direction. Label each of the n crossings with the numbers 1, ..., 2''n'' in order of traversal (each crossing is visited and labelled twice), with the following modification: if the label is an even number and the strand followed crosses over at the crossing, then change the sign on the label to be a negative. When finished, each crossing will be labelled a pair of integers, one even and one odd. The Dowker–Thistlethwaite notation is the sequence of even integer labels associated with the labels 1, 3, ..., 2''n'' − 1 in turn.

Example

For example, a

knot diagram 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 k ...

may have crossings labelled with the pairs (1, 6) (3, −12) (5, 2) (7, 8) (9, −4) and (11, −10). The Dowker–Thistlethwaite notation for this labelling is the sequence: 6 −12 2 8 −4 −10.

Uniqueness and counting

Dowker and Thistlethwaite have proved that the notation specifies

prime knot In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non- trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be ...

s uniquely,

up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech ...

reflection. In the more general case, a knot can be recovered from a Dowker–Thistlethwaite sequence, but the recovered knot may differ from the original by either being a reflection or by having any

connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...

component reflected in the line between its entry/exit points – the Dowker–Thistlethwaite notation is unchanged by these reflections. Knots tabulations typically consider only

s and disregard

chirality Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable fro ...

, so this ambiguity does not affect the tabulation. The ménage problem, posed by Tait, concerns counting the number of different number sequences possible in this notation.

References

External links

*
DT Notation
''Knotinfo''

{{DEFAULTSORT:Dowker-Thistlethwaite notation Knot theory Mathematical notation

Definition

Example

Uniqueness and counting

See also

References

Further reading

External links