Perko pair


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 Tait-Little knot tables of knots up to 10 crossings, 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, as the two diagrams for the pair have different writhes.
In some later knot tables, the knots have been renumbered slightly 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.