In the
mathematical theory of knots, the Perko pair, named after Kenneth Perko, is a pair of entries in classical knot tables that actually represent the same knot. In
Dale Rolfsen
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.
Table of prime kn ...
's knot table, this supposed pair of distinct knots is labeled 10
161 and 10
162. In 1973, while working to complete the classification by knot type of the Tait–Little knot tables of knots up to 10 crossings (dating from the late 19th century), Perko found the duplication in
Charles Newton Little
Charles Newton Little (1858–1923) was an American mathematician and civil engineer. He was known for his expertise in knot theory, including the construction of a table of knots with ten or fewer crossings...
Little's father was a missionary ...
's table. This duplication had been missed 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 branches ...
several years before in his knot table and subsequently found its way into Rolfsen's table. The Perko pair gives a counterexample to a "theorem" claimed by Little in 1900 that the
writhe
In knot theory, there are several competing notions of the quantity writhe, or \operatorname. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amou ...
of a reduced diagram of a knot is an invariant (see
Tait conjectures), as the two diagrams for the pair have different writhes.
In some later knot tables, the knots have been renumbered slightly (knots 10
163 to 10
166 are renumbered as 10
162 to 10
165) so that knots 10
161 and 10
162 are different. Some authors have mistaken these two renumbered knots for the Perko pair and claimed incorrectly that they are the same.
The Revenge of the Perko Pair
, ''RichardElwes.co.uk''. Accessed February 2016. Richard Elwes points out a common mistake in describing the Perko pair.
The Perko pair was correctly illustrated and explained on the first page of the Science section of the July 8, 1986 New York Times.
References
External links
*
*Pictures of the equivalence between the two knots:
Perko pair knots, ''KnotPlot''. Accessed February 2016.
{{knot theory