Perko pair

{{Short description|Prime knot with crossing number 10}}

{{Infobox knot theory

| name= Perko pair

| practical name=

| image=

| caption=

| arf invariant= 1

| braid length= 10

| braid number= 3

| bridge number= 3

| crosscap number= 2

| crossing number= 10

| genus= 3

| hyperbolic volume= 5.63877

| linking number=

| stick number=

| unknotting number= 3

| conway_notation= [3:-20:-20]

| ab_notation= 10₁₆₁/10₁₆₂

| dowker notation= 4, 12, -16, 14, -18, 2, 8, -20, -10, -6

| thistlethwaite=

| last crossing= 10

| last order= 160

| next crossing= 10

| next order= 162

| alternating=

| class= hyperbolic

| fibered= fibered

| pretzel=

| prime= prime

| slice=

| symmetry= reversible

| tricolorable=

| twist=

}}

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),Charles Newton Little, Non-alternating +/- knots, Trans. Roy. Soc. Edinburgh 39 (1900), page 774. Perko found the duplication in Charles Newton Little's table.Kenneth A. Perko Jr.(b.1943), [https://www.ams.org/journals/proc/1974-045-02/S0002-9939-1974-0353294-X/S0002-9939-1974-0353294-X.pdf On the classification of knots.] Proc. Amer. Math. Soc. 45 (1974), 262—266. 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.Dale Rolfsen, Knots and Links (see Appendix C for the knot table), 1976, {{isbn|0-914098-16-0}}. 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."[http://richardelwes.co.uk/2013/08/14/the-revenge-of-the-perko-pair/ The Revenge of the Perko Pair]", RichardElwes.co.uk. Accessed February 2016. Richard Elwes points out a common mistake in describing the Perko pair.

Image:Ten onehundredandsixtyone.gif|{{center|10₁₆₁}}

Image:Ten onehundredandsixtytwo.gif|{{center|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.