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Almost Alternating Link

Call a projection of a Link an almost alternating projection if one crossing change in the projection makes it an alternating projection. Then an almost alternating link is a Link with an almost alternating projection, but no alternating projection. Every Alternating Knot has an almost alternating projection. A Prime Knot which is almost alternating is either a Torus Knot or a Hyperbolic Knot. Therefore, no Satellite Knot is an almost alternating knot.

All nonalternating 9-crossing Prime Knots are almost alternating. Of the 393 nonalternating knots and links with 11 or fewer crossings, all but five are known to be almost alternating (3 of these have 11 crossings). The fate of the remaining five is not known. The $(2,q)$, $(3,4)$, and $(3,5)$-Torus Knots are almost alternating.

See also Alternating Knot, Link


Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 139-146, 1994.

© 1996-9 Eric W. Weisstein