Alternating Knot

An alternating knot is a knot which possesses a knot diagram in which crossings alternate between under- and overpasses. Not all knot diagrams of alternating knots need be alternating diagrams.

The trefoil knot and figure eight knot are alternating knots, as are all prime knots with seven or fewer crossings. A knot can be checked in the Wolfram Language to see if it is alternating using KnotData[knot, "Alternating"].

The number of prime alternating and nonalternating knots of crossings are summarized in the following table.

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).

If has a reduced alternating projection of crossings, then the link span of is . Let be the link crossing number. Then an alternating knot (a knot sum) satisfies

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.