Perko pair

Perko pair
Arf invariant	1
Braid length	10
Braid no.	3
Bridge no.	3
Crosscap no.	2
Crossing no.	10
Genus	3
Hyperbolic volume	5.63877
Unknotting no.	3
Conway notation	[3:-20:-20]
A–B notation	10161/10162
Dowker notation	4, 12, -16, 14, -18, 2, 8, -20, -10, -6
Last / Next	10160 / 10162
Other
hyperbolic, fibered, prime, reversible

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's knot table, this supposed pair of distinct knots is labeled 10₁₆₁ and 10₁₆₂. 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's table. This duplication had been missed by John Horton Conway 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 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 10₁₆₃ to 10₁₆₆ are renumbered as 10₁₆₂ to 10₁₆₅) so that knots 10₁₆₁ and 10₁₆₂ are different. Some authors have mistaken these two renumbered knots for the Perko pair and claimed incorrectly that they are the same.

• The Perko pair
• 
  10₁₆₁
• 
  10₁₆₂ (in Rolfsen's original numbering)

The Perko pair was correctly illustrated and explained on the first page of the Science section of the July 8, 1986 New York Times.

External links

• "10_161", The Knot Atlas.
• Pictures of the equivalence between the two knots, as given by Perko: "The Perko pair", hosted by Brian Sanderson. Accessed January 2023.
• Pictures of a different equivalence between the two knots: "Perko pair knots", KnotPlot. Accessed February 2016.