A nonalternating knot is 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 at least 13 different knot moves in generating an initial list of 16-crossing nonalternating knots, which still contained 9868 duplicates among its 1018774 entries (Hoste et al. 1998).
The numbers of nonalternating prime knots with 
, 2, ... crossings, counting a knot
and its mirror image only once, are 0, 0, 0, 0, 0,
0, 0, 3, 8, 42, 185, 888, ... (OEIS A051763),
the first few of which are 
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Knots." Math. Intell. 20, 33-48, 1998.