Alternating knot

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One of three non-alternating knots with crossing number 8

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

Many of the knots with crossing number less than 10 are alternating. This fact and useful properties of alternating knots, such as the Tait conjectures, was what enabled early knot tabulators, such as Tait, to construct tables with relatively few mistakes or omissions. 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.

Alternating links end up having an important role in knot theory and 3-manifold theory, due to their complements having useful and interesting geometric and topological properties. This led Ralph Fox to ask, "What is an alternating knot?" By this he was asking what non-diagrammatic properties of the knot complement would characterize alternating knots.

Various geometric and topological information is revealed in an alternating diagram. Primeness and splittability of a link is easily seen from the diagram. Applying Seifert's algorithm to a knot results in a minimal genus Seifert surface. Crossing number of a reduced, alternating diagram is the crossing number of the knot. This last is one of the celebrated Tait Conjectures.

An alternating knot diagram is in one to one correspondence with a planar graph. Each crossing is associated with an edge and half of the connected components of the complement of the diagram are associated with vertices in a checker board manner.

1 Tait conjectures
- 1.1 Flypes
2 Hyperbolic volume
3 References
4 External links

[edit] Tait conjectures

Any two reduced diagrams of the same alternating knot have the same writhe. This fact, one of the Tait conjectures, was proven 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 William Menasco and Morwen Thistlethwaite in 1991 and 1993.

[edit] Flypes

[edit] Hyperbolic volume

Menasco, applying Thurston's geometrization theorem for Haken manifolds, showed that any prime, non-split alternating link is hyperbolic, i.e. the link complement has a hyperbolic geometry, unless the link is a torus link.

Thus hyperbolic volume is an invariant of many alternating links. Marc Lackenby has shown that the volume has upper and lower linear bounds as functions of the number of twist regions of a reduced, alternating diagram.

[edit] References

Louis H. KauffmanOn Knots, Princeton University Press, 1987. ISBN 0-691-08435-1.
Colin Adams, The Knot Book, American Mathematical Society, ISBN 0-8050-7380-9
William Menasco, Closed incompressible surfaces in alternating knot and link complements. Topology 23 (1984), no. 1, 37--44.
Marc Lackenby, The volume of hyperbolic alternating link complements. With an appendix by Ian Agol and Dylan Thurston. Proc. London Math. Soc. (3) 88 (2004), no. 1, 204--224.