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Nonalternating Knot


A knot which is not alternating. Unlike alternating knots, flype moves are not sufficient to pass between all minimal diagrams of a given nonalternating knot (Hoste et al. 1998). In fact, Thistlethwaite used 13 different moves in generating a list of 16-crossing alternating knots (Hoste et al. 1998), and still had 9868 duplicates out of a list of 1018774 knots (Hoste et al. 1998).

The numbers of nonalternating knots with n=1, 2, ... crossings are 0, 0, 0, 0, 0, 0, 0, 3, 8, 42, 185, 888, ... (OEIS A051763), the first few of which are 8_(19), 8_(20), 8_(21), 9_(42), 9_(43), 9_(44), 9_(45), 9_(46), 9_(47), 9_(48), 9_(49), 10_(124), 10_(125), 10_(126), 10_(127), 10_(128), 10_(129), 10_(130), 10_(131), 10_(132), 10_(133), 10_(134), 10_(135), 10_(136), 10_(137), 10_(138), 10_(139), 10_(140), 10_(141), 10_(142), 10_(143), 10_(144), 10_(145), 10_(146), 10_(147), 10_(148), 10_(149), 10_(150), 10_(151), 10_(152), 10_(153), 10_(154), 10_(155), 10_(156), 10_(157), 10_(158), 10_(159), 10_(160), 10_(161), 10_(162), 10_(163), 10_(164), and 10_(165).


See also

Alternating Knot, Knot

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References

Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1701936 Knots." Math. Intell. 20, 33-48, Fall 1998.Sloane, N. J. A. Sequence A051763 in "The On-Line Encyclopedia of Integer Sequences."

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Nonalternating Knot

Cite this as:

Weisstein, Eric W. "Nonalternating Knot." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NonalternatingKnot.html

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