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 (and 3 of these have 11 crossings). The fate of the remaining five is not known. The -, -, and -torus knots are almost alternating (Adams 1994, p. 142).

REFERENCES:

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