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[[Image:Crossing numbers trefoil.png|thumb|Trefoil knot without 3-fold symmetry with crossings labeled.]]
[[Image:Knot table.svg|thumb|A table of all [[prime knot]]s with seven crossing numbers or fewer (not including mirror images).]]

In the [[mathematics|mathematical]] area of [[knot theory]], the '''crossing number''' of a [[knot (mathematics)|knot]] is the smallest number of crossings of any diagram of the knot. It is a [[knot invariant]].

==Examples==
By way of example, the [[unknot]] has crossing number [[0 (number)|zero]], the [[trefoil knot]] three and the [[figure-eight knot (mathematics)|figure-eight knot]] four. There are no other knots with a crossing number this low, and just two knots have crossing number five, but the number of knots with a particular crossing number increases rapidly as the crossing number increases.

==Tabulation==
Tables of [[prime knot]]s are traditionally indexed by crossing number, with a subscript to indicate which particular knot out of those with this many crossings is meant (this sub-ordering is not based on anything in particular, except that [[torus knot]]s then [[twist knot]]s are listed first). The listing goes 3<sub>1</sub> (the trefoil knot), 4<sub>1</sub> (the figure-eight knot), 5<sub>1</sub>, 5<sub>2</sub>, 6<sub>1</sub>, etc. This order has not changed significantly since [[P. G. Tait]] published a tabulation of knots in 1877.<ref>{{citation|authorlink=P.G. Tait|last=Tait|first=P. G.|contribution=On Knots I,II,III'|title=Scientific papers|volume=1|pages=273–347|publisher=Cambridge University Press|year=1898}}.</ref>

==Additivity==
There has been very little progress on understanding the behavior of crossing number under rudimentary operations on knots. A big open question asks if the crossing number is additive when taking [[Connected sum#Connected sum of knots|knot sums]]. It is also expected that a [[satellite knot|satellite]] of a knot ''K'' should have larger crossing number than ''K'', but this has not been proven.

Additivity of crossing number under knot sum has been proven for special cases, for example if the summands are [[alternating knot]]s<ref>{{citation|authorlink=Colin Adams (mathematician)|last=Adams|first=C. C.|year=2004|title=The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots|publisher=American Mathematical Society|isbn= 9780821836781|page=69|url=http://books.google.com/books?id=M-B8XedeL9sC&pg=PA69}}.</ref> (or more generally, [[adequate knots|adequate knot]]), or if the summands are [[torus knot]]s.<ref>{{citation|last=Gruber|first=H.|title=Estimates for the minimal crossing number|arxiv=math/0303273|year=2003}}.</ref><ref>{{citation
 | last = Diao | first = Yuanan
 | doi = 10.1142/S0218216504003524
 | issue = 7
 | journal = Journal of Knot Theory and its Ramifications
 | mr = 2101230
 | pages = 857–866
 | title = The additivity of crossing numbers
 | volume = 13
 | year = 2004}}.</ref>  Marc Lackenby has also given a proof that there is a constant ''N''&nbsp;>&nbsp;1 such that <math>\frac{1}{N} (\mathrm{cr}(K_1) + \mathrm{cr}(K_2)) \leq \mathrm{cr}(K_1 + K_2)</math>, but his method, which utilizes [[normal surface]]s, cannot improve ''N'' to 1.<ref>{{citation
 | last = Lackenby | first = Marc
 | doi = 10.1112/jtopol/jtp028
 | issue = 4
 | journal = Journal of Topology
 | mr = 2574742
 | pages = 747–768
 | title = The crossing number of composite knots
 | url = http://www.maths.ox.ac.uk/~lackenby/csk16058.ps
 | volume = 2
 | year = 2009}}.</ref>

==Applications in bioinformatics==
There are mysterious connections between the crossing number of a knot and the physical behavior of [[DNA]] knots. For prime DNA knots, crossing number is a good predictor of the relative velocity of the DNA knot in agarose [[gel electrophoresis]]. Basically, the higher the crossing number, the faster the relative velocity. For composite knots, this does not appear to be the case, although experimental conditions can drastically change the results.<ref>{{citation|contribution=Energy functions for knots: Beginning to predict physical behavior|title=Mathematical Approaches to Biomolecular Structure and Dynamics|series=The IMA Volumes in Mathematics and its Applications|volume=82|year=1996|pages=39–58|first=Jonathan|last=Simon|doi=10.1007/978-1-4612-4066-2_4|editor1-first=Jill P.|editor1-last=Mesirov|editor2-first=Klaus|editor2-last=Schulten|editor3-first=De Witt|editor3-last=Sumners}}.</ref>

==Related invariants==
There are related concepts of [[average crossing number]] and [[asymptotic crossing number]]. Both of these quantities bound the standard crossing number.  Asymptotic crossing number is conjectured to be equal to crossing number.

Other numerical knot invariants include the [[bridge number]], [[linking number]], [[stick number]], and [[unknotting number]].

==References==
<references/>

{{Knot theory|state=collapsed}}

{{DEFAULTSORT:Crossing Number (Knot Theory)}}
[[Category:Knot invariants]]