Editing Alternating knot

[[Image:Knot 8sb19.svg|right|thumb|250px|One of three non-alternating knots with [[crossing number (knot theory)|crossing number]] 8]]

In [[knot theory]], a [[knot (mathematics)|knot]] or [[link (knot theory)|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 (knot theory)|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 knot]]s have 8 crossings (and there are three such: 8<sub>19</sub>, 8<sub>20</sub>, 8<sub>21</sub>).

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 [[knot complement|complement]]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.<ref>{{citation
 | last = Lickorish | first = W. B. Raymond
 | chapter = Geometry of Alternating Links
 | doi = 10.1007/978-1-4612-0691-0_4
 | isbn = 0-387-98254-X
 | mr = 1472978
 | pages = 32–40
 | publisher = Springer-Verlag, New York
 | series = Graduate Texts in Mathematics
 | title = An Introduction to Knot Theory
 | volume = 175
 | year = 1997}}; see in particular [https://books.google.com/books?id=xSLUBwAAQBAJ&pg=PA32 p. 32]</ref>

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]].<ref>{{cite journal|last1=Greene|first1=Joshua|title=Alternating links and definite surfaces|journal=Duke Mathematical Journal|year=2017|volume=166|issue=11|doi=10.1215/00127094-2017-0004|arxiv=1511.06329|s2cid=59023367}}</ref>

Various geometric and topological information is revealed in an alternating diagram. Primeness and [[split link|splittability]] of a link is easily seen from the diagram. The crossing number of a [[reduced diagram|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.

[[Image:Trefle.jpg]]

[[Image:Frise.jpg]]

==Tait conjectures==
{{main article|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]].
#Given any two reduced alternating diagrams D<sub>1</sub> and D<sub>2</sub> of an oriented, prime alternating link: D<sub>1</sub> may be transformed to D<sub>2</sub> by means of a sequence of certain simple moves called ''[[flype]]s''. Also known as the Tait flyping conjecture.<ref>{{MathWorld|title=Tait's Knot Conjectures|urlname=TaitsKnotConjectures}} Accessed: May 5, 2013.</ref> 
[[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==

[[William Menasco|Menasco]], applying [[William Thurston|Thurston]]'s [[hyperbolization theorem]] for [[Haken manifold]]s, showed that any prime, non-split alternating link is [[hyperbolic link|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.

==References==
{{reflist}}

==Further reading==
{{refbegin}}
* {{cite book | first=Louis H. | last=Kauffman | authorlink = Louis Kauffman | title=On Knots | zbl=0627.57002 | series=Annals of Mathematics Studies | volume=115 | publisher=Princeton University Press | year=1987 | isbn=0-691-08435-1 }}
*{{cite book |author-link=Colin Adams (mathematician) |first=Colin C. |last=Adams |title=The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots |url=https://books.google.com/books?id=RqiMCgAAQBAJ |year=2004 |publisher=American Mathematical Society |isbn=978-0-8218-3678-1 }}
*{{cite journal |author-link=William Menasco |first=William  |last=Menasco |title=Closed incompressible surfaces in alternating knot and link complements |journal=Topology |volume=23 |issue=1 |pages=37–44 |year=1984 |doi=10.1016/0040-9383(84)90023-5 |url=https://core.ac.uk/download/pdf/81997602.pdf|doi-access=free }}
*{{cite journal |author-link=Marc Lackenby |first=Marc  |last=Lackenby |title=The volume of hyperbolic alternating link complements |journal=Proc. London Math. Soc. |volume=88 |issue=1 |pages=204–224 |year=2004 |doi=10.1112/S0024611503014291 |url= |arxiv=math/0012185|s2cid=56284382 }}
{{refend}}

==External links==

*{{MathWorld|title=Alternating Knot|id=AlternatingKnot}}
*{{MathWorld|title=Tait's Knot Conjectures|id=TaitsKnotConjectures}}
*[http://www.entrelacs.net Celtic Knotwork] to build an alternating knot from its planar graph

{{Knot theory|state=collapsed}}

[[Category:Alternating knots and links| ]]
[[Category:Knot invariants]]