In knot theory, a knot diagram is alternating if the crossings alternate under, over, under, over, as you travel along the strand. A link diagram is alternating if each strand has this property. A knot or link is alternating if it has an alternating diagram.

The simplest non-alternating prime knots have 8 crossings (and there are three such).

It is conjectured that as the crossing number increases, the percentage of knots that are alternating goes to 0 exponentially quickly.

Reduced diagrams

Reduced diagrams of alternating knots are particularly nice.

Any two reduced diagrams of the same alternating knot have the same writhe. This fact was first conjectured by P. G. Tait, and proved by Louis Kauffman , K. Murasugi , and Morwen Thistlethwaite in 1987 and 1988.

Also, any reduced diagram of an alternating knot or link has the fewest possible crossings. This fact follows from the Tait flyping conjecture , which was proved by W. Menasco and Thistlethwaite in 1991 and 1993.

References

• On Knots, Louis H. Kauffman, Princeton University Press, 1987. ISBN 0-691-08435-1.

External links

• Alternating Knot at MathWorld
• Tait's Knot Conjectures at MathWorld