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] 
AB notation  10_{161}/10_{162} 
Dowker notation  4, 12, 16, 14, 18, 2, 8, 20, 10, 6 
Last /Next  10<sub>160</sub> / 10<sub>162</sub> 
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 10_{161} and 10_{162}. In 1973, while working to complete the Tait–Little knot tables of knots up to 10 crossings (dating from the late 19th century),^{[1]} Perko found the duplication in Charles Newton Little's table.^{[2]} 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.^{[3]} 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 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.^{[4]}
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Chapter 3 exercise 3.5 Q4 (all parts) pair of linear equations in two variables maths class 10
Transcription
References
 ^ Charles Newton Little, Nonalternating +/ knots, Trans. Roy. Soc. Edinburgh 39 (1900), page 774.
 ^ Kenneth A. Perko Jr.(b.1943), On the classification of knots. Proc. Amer. Math. Soc. 45 (1974), 262—266.
 ^ Dale Rolfsen, Knots and Links (see Appendix C for the knot table), 1976, ISBN 0914098160.
 ^ "The Revenge of the Perko Pair", RichardElwes.co.uk. Accessed February 2016. Richard Elwes points out a common mistake in describing the Perko pair.
External links
 "10_161", The Knot Atlas.
 Pictures of the equivalence between the two knots: "Perko pair knots", KnotPlot. Accessed February 2016.