In
knot theory, a
knot
A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...
or
link diagram is alternating if the crossings alternate under, over, under, over, as one travels 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: 8
19, 8
20, 8
21).
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
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
s 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.
In November 2015, Joshua Evan Greene published a preprint that established a characterization of alternating links in terms of definite spanning surfaces, i.e. a definition of alternating links (of which alternating knots are a special case) without using the concept of a
link diagram.
Various geometric and topological information is revealed in an alternating diagram. Primeness and
splittability of a link is easily seen from the diagram. The 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
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cro ...
. 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.
Tait conjectures
The Tait conjectures are:
#Any reduced diagram of an alternating link has the fewest possible crossings.
#Any two reduced diagrams of the same alternating knot have the same
writhe
In knot theory, there are several competing notions of the quantity writhe, or \operatorname. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amoun ...
.
#Given any two reduced alternating diagrams D
1 and D
2 of an oriented, prime alternating link: D
1 may be transformed to D
2 by means of a sequence of certain simple moves called ''
flypes''. Also known as the Tait flyping conjecture.
[ Accessed: May 5, 2013.]
Morwen Thistlethwaite,
Louis Kauffman and
K. Murasugi proved the first two Tait conjectures in 1987 and
Morwen Thistlethwaite and
William Menasco proved the Tait flyping conjecture in 1991.
Hyperbolic volume
Menasco, applying
Thurston's
hyperbolization theorem for
Haken manifolds, showed that any prime, non-split alternating link is
hyperbolic, i.e. the link complement has a
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P' ...
, 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.
References
Further reading
*
*
*
*
External links
*
*
Celtic Knotworkto build an alternating knot from its planar graph
{{Knot theory, state=collapsed
Knot invariants