Editing Alternating knot (section)

[[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.

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