The 3 nonalternating knots of eight crossings are , , and , illustrated above (Wells 1991).
One of Tait's knot conjectures states that the number of crossings is the same for any diagram of a reduced alternating knot.
Furthermore, a reduced alternating projection of a knot has the least number of crossings
for any projection of that knot. Both of these facts were proved true by Kauffman
(1988), Thistlethwaite (1987), and Murasugi (1987). Flype
moves are sufficient to pass between all minimal diagrams of a given alternating
knot (Hoste et al. 1998).
In fact, this is true as well for the larger class of adequate
knots and postulated for all knots.
It is conjectured that the proportion of knots which are alternating tends exponentially to zero with increasing crossing number (Hoste et al. 1998), a statement which
has been proved true for alternating links.
Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots.
New York: W. H. Freeman, pp. 159-164, 1994.Arnold, B.; Au,
M.; Candy, C.; Erdener, K.; Fan, J.; Flynn, R.; Muir, J.; Wu, D.; and Hoste, J. "Tabulating
Alternating Knots through 14 Crossings." ftp://chs.cusd.claremont.edu/pub/knot/paper.TeX.txt.Arnold,
B.; Au, M.; Candy, C.; Erdener, K.; Fan, J.; Flynn, R.; Muir, J.; Wu, D.; and Hoste,
K. and Flynn, R. "Rolfsen's Table of all Alternating Diagrams through 9 Crossings."
J.; Thistlethwaite, M.; and Weeks, J. "The First Knots." Math. Intell.20, 33-48,
Fall 1998.Kauffman, L. "New Invariants in the Theory of Knots."
Amer. Math. Monthly95, 195-242, 1988.Little, C. N.
Knots of Orders Eight and Nine." Trans. Roy. Soc. Edinburgh35,
663-664, 1889.Little, C. N. "Alternate Knots of Order 11." Trans. Roy. Soc. Edinburgh36,
253-255, 1890.Little, C. N. "Non-Alternate Knots." Trans. Roy. Soc. Edinburgh39,
771-778, 1900.Livingston, C. Knot
Theory. Washington, DC: Math. Assoc. Amer., pp. 6, 132, and 219, 1993.Murasugi,
K. "Jones Polynomials and Classical Conjectures in Knot Theory." Topology26,
297-307, 1987.Sloane, N. J. A. Sequences A002864/M0847
and A051763 in "The On-Line Encyclopedia
of Integer Sequences."Thistlethwaite, M. "A Spanning Tree
Expansion for the Jones Polynomial." Topology26, 297-309, 1987.Wells,
Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
p. 160, 1991.