View source for Reidemeister move

{{short description|One of three types of isotopy-preserving local changes to a knot diagram}}
{| class="infobox" style="background:#fff"
|+ Reidemeister moves
|- style="padding:1em"
| [[Image:Reidemeister move 1.svg|130px]]
| [[Image:Reidemeister move 2.svg|130px]]
| [[Image:Reidemeister move 3.svg|130px]]
|-
| Type I || Type II || Type III
|}

{| class="infobox" style="background:#fff"
|+ Modified Reidemeister move
|- style="padding:1em"
| [[Image:Reidemeister move 1 prime.svg|130px]]
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| Type I'
|}

In the [[mathematics|mathematical]] area of [[knot theory]], a '''Reidemeister move''' is any of three local moves on a [[knot diagram|link diagram]]. {{harvs|txt|last=Reidemeister|first=Kurt|authorlink=Kurt Reidemeister|year=1927}} and, independently, {{harvs|txt|first1=James Waddell |last1=Alexander| authorlink1=James Waddell Alexander II | first2=Garland Baird | last2=Briggs | year=1926|authorlink2=Garland Baird Briggs}}, demonstrated that two knot diagrams belonging to the same knot, up to planar [[regular isotopy|isotopy]], can be related by a sequence of the three Reidemeister moves.{{Sfnp|Reidemeister|1927|pp=24–25}}{{Sfnp|Alexander|Briggs|1926|p=562}}

Each move operates on a small region of the diagram and is one of three types:{{Sfnp|Reidemeister|1927|p=25}}
{{Ordered list|list-style-type=upper-roman
| Twist and untwist in either direction.
| Move one loop completely over another.
| Move a string completely over or under a crossing.
}}

No other part of the diagram is involved in the picture of a move, and a planar isotopy may distort the picture. The numbering for the types of moves corresponds to how many strands are involved, e.g. a type II move operates on two strands of the diagram.

One important context in which the Reidemeister moves appear is in defining [[knot invariants]].{{Sfnp|Östlund|2001|p=1219}} By demonstrating a property of a knot diagram which is not changed when we apply any of the Reidemeister moves, an invariant is defined.  Many important invariants can be defined in this way, including the [[Jones polynomial]].

The type I move is the only move that affects the [[writhe]] of the diagram. The type III move is the only one which does not change the crossing number of the diagram.{{Sfnp|Trace|1983|p=723}}

In applications such as the [[Kirby calculus]], in which the desired [[equivalence class]] of knot diagrams is not a knot but a [[framed link]], one must replace the type I move with a "modified type I" (type I') move composed of two type I moves of opposite sense.  The type I' move affects neither the framing of the link nor the writhe of the overall knot diagram.{{Sfnp|Alexander|Briggs|1926|p=564}}

{{harvtxt|Trace|1983}} showed that two knot diagrams for the same knot are related by using only type II and III moves if and only if they have the same [[writhe]] and [[winding number]].{{Sfnp|Trace|1983|p=722}} Furthermore, combined work of {{harvtxt|Östlund|2001}}, {{harvtxt|Manturov|2004}}, and {{harvtxt|Hagge|2006}} shows that for every knot type there are a pair of knot diagrams so that every sequence of Reidemeister moves taking one to the other must use all three types of moves.{{Sfnp|Östlund|2001|pp=1215-1227}} Alexander Coward demonstrated that for link diagrams representing equivalent links, there is a sequence of moves ordered by type: first type I moves, then type II moves, type III, and then type II. The moves before the type III moves increase crossing number while those after decrease crossing number.

{{harvtxt|Coward|Lackenby|2014}} proved the existence of an exponential tower [[upper bound]] (depending on crossing number) on the number of Reidemeister moves required to pass between two diagrams of the same link.{{Sfnp|Coward|Lackenby|2014|p=1024}} In detail, let <math>n</math> be the sum of the crossing numbers of the two diagrams, then the upper bound is <math>2^{2^{2^{.^{.^n}}}}</math> where the height of the tower of <math>2</math>s (with a single <math>n</math> at the top) is <math>10^{1,000,000n}</math>.

{{harvtxt|Lackenby|2015}} proved the existence of a polynomial upper bound (depending on crossing number) on the number of Reidemeister moves required to change a diagram of the unknot to the standard unknot. In detail, for any such diagram with <math>c</math> crossings, the upper bound is <math>(236 c)^{11}</math>.{{Sfnp|Lackenby|2015|p=1}}

{{harvtxt|Hayashi|2005}} proved there is also an upper bound, depending on crossing number, on the number of Reidemeister moves required to [[split link|split a link]].{{Sfnp|Hayashi|2005|p=1}}

==References==
{{Reference}}
*{{commons category inline|Reidemeister moves}}
* {{citation | first1=James W. | last1=Alexander | first2=Garland B. | last2=Briggs | doi=10.2307/1968399 | title=On types of knotted curves | journal=[[Annals of Mathematics]] | volume=28 | year=1926 | issue=1/4 | pages=562–586 | jstor=1968399 | mr=1502807 }}
*{{citation|last1=Coward|first1=Alexander|last2=Lackenby|first2=Marc|author2-link=Marc Lackenby|title=An upper bound on Reidemeister moves|journal=[[American Journal of Mathematics]]|year=2014|volume=136|issue=4|pages=1023–1066|doi=10.1353/ajm.2014.0027|url=https://muse.jhu.edu/journals/american_journal_of_mathematics/summary/v136/136.4.coward.html|mr=3245186|arxiv=1104.1882|s2cid=55882290}}
* {{citation | first=Stefano | last=Galatolo | title=On a problem in effective knot theory | journal=Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. | volume=9 | issue=4 | pages=299–306 | year=1999 | mr=1722788 | url=https://eudml.org/doc/252452}}
* {{citation | first=Tobias | last=Hagge | title=Every Reidemeister move is needed for each knot type | journal=[[Proc. Amer. Math. Soc.]] | volume=134 | year=2006 | issue=1 | pages=295–301 | mr=2170571 | doi=10.1090/S0002-9939-05-07935-9 | doi-access=free }}
*{{citation
 | last1 = Hass | first1 = Joel | author1-link = Joel Hass
 | last2 = Lagarias | first2 = Jeffrey C. | author2-link = Jeffrey Lagarias
 | doi = 10.1090/S0894-0347-01-00358-7
 | issue = 2
 | journal = [[Journal of the American Mathematical Society]]
 | mr = 1815217
 | pages = 399–428
 | title = The number of Reidemeister moves needed for unknotting
 | volume = 14
 | year = 2001| arxiv = math/9807012| s2cid = 15654705 }}
*{{citation | first=Chuichiro | last=Hayashi | title=The number of Reidemeister moves for splitting a link | journal=[[Mathematische Annalen]] | volume=332 | year=2005 | pages=239–252 | issue=2 | mr=2178061 | doi=10.1007/s00208-004-0599-x | s2cid=119728321 }}
*{{citation
 | last = Lackenby | first = Marc | author-link = Marc Lackenby
 | arxiv = 1302.0180
 | doi = 10.4007/annals.2015.182.2.3
 | issue = 2
 | journal = [[Annals of Mathematics]]
 | mr = 3418524
 | pages = 491–564
 | series = Second Series
 | title = A polynomial upper bound on Reidemeister moves
 | volume = 182
 | year = 2015| s2cid = 119662237 }}
*{{citation | first=Vassily Olegovich | last=Manturov | title=Knot theory | isbn=0-415-31001-6 | year=2004 | mr=2068425 | doi=10.1201/9780203402849 | publisher=Chapman & Hall/CRC | location=Boca Raton, FL }}
*{{citation | first=Olof-Petter | last=Östlund | title=Invariants of knot diagrams and relations among Reidemeister moves | journal= J. Knot Theory Ramifications | volume=10 | issue=8 | year=2001 | pages=1215–1227 | mr=1871226 | doi=10.1142/S0218216501001402 | arxiv=math/0005108 | s2cid=119177881 }}
* {{citation
 | last1 = Reidemeister | first1 = Kurt
 | authorlink1 = Kurt Reidemeister
 | title = Elementare Begründung der Knotentheorie
 | journal = Abh. Math. Sem. Univ. Hamburg
 | volume = 5
 | issue = 1
 | pages = 24–32
 | year = 1927
 | doi = 10.1007/BF02952507
 | mr = 3069462| s2cid = 120149796
 }}
* {{citation | first=Bruce | last=Trace | title=On the Reidemeister moves of a classical knot | journal=[[Proceedings of the American Mathematical Society]] | volume=89 | year=1983 | issue=4 | pages=722–724 | mr=0719004 | doi=10.2307/2044613 | jstor=2044613 | doi-access=free }}

{{Knot theory|state=collapsed}}

[[Category:Knot operations]]