Perko Pair

The Perko pair consists of the two knot diagrams labeled and in the original numbering of Rolfsen (1976).

The original knot diagrams from Rolfsen (1976, p. 415) are shown above with their original labels and handedness (cf. Bar-Natan).

For many years, they were listed as separate knots in Little (1900) and all similar tables, including the pictorial enumeration of Rolfsen (1976, Appendix C). Perko discovered their equivalence in 1973 (Bar-Natan) and published the result in 1974 (Perko 1974). They are related to one another by the so-called Perko move (Perko 1974, Hoste et al. 1998). Rolfsen's 1976 first edition displays both and without noting their equivalence (Rolfsen 1976, p. 415). The corrected 2003 printing retains the same pagination and both diagrams, but adds a parenthetical note directly beneath that it is equivalent to (Rolfsen 2003, p. 415). Although these knots are equivalent, their knot diagrams have different writhes (Hoste et al. 1998).

The redrawn figure above reflects the left knot diagram so that both knot diagrams have the same handedness.

In the Knot Atlas, the duplicate is removed and the original through are renumbered as through , reducing the total from 166 listed ten-crossing knots to 165 distinct ten-crossing knots (Bar-Natan). Consequently, and in that numbering are distinct knots and do not form the Perko pair.