In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of knot theory, the Dowker–Thistlethwaite (DT) notation or code, for a

knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

is a sequence of even

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

s. The notation is named after

Clifford Hugh Dowker Clifford Hugh Dowker (; March 2, 1912 – October 14, 1982) was a topologist known for his work in point-set topology and also for his contributions in category theory, sheaf theory and knot theory. Biography Clifford Hugh Dowker grew up on a sm ...

and

Morwen Thistlethwaite Morwen Bernard Thistlethwaite is a knot theorist and professor of mathematics for the University of Tennessee in Knoxville. He has made important contributions to both knot theory and Rubik's Cube group theory. Biography Morwen Thistlethwait ...

, who refined a notation originally due to

Peter Guthrie Tait Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook ''Treatise on Natural Philosophy'', which he co-wrote wi ...

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 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 co ...

s uniquely, up to 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, so this ambiguity does not affect the tabulation. The

ménage problem In combinatorial mathematics, the ménage problem or problème des ménages asks for the number of different ways in which it is possible to seat a set of male-female couples at a round dining table so that men and women alternate and nobody sits ...

, 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