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[[File:Reidemeister moves.jpg|thumb|Diagram of Reidemeister moves.]]
[[File:Reidemeister moves.jpg|thumb|Diagram of Reidemeister moves.]]
In the mathematical area of [[Knot theory (nonfiction)|knot theory]], a '''Reidemeister move''' is any of three local moves on a link diagram. [[Kurt Reidemeister (nonfiction)|Kurt Reidemeister]] (1927) and, independently, James Waddell Alexander and Garland Baird Briggs (1926), demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves.
In the mathematical area of [[Knot theory (nonfiction)|knot theory]], a '''Reidemeister move''' is any of three local moves on a link diagram. [[Kurt Reidemeister (nonfiction)|Kurt Reidemeister]] (1927) and, independently, [[James Waddell Alexander II (nonfiction)|James Waddell]] Alexander and [[Garland Baird Briggs (nonfiction)|Garland Baird Briggs]] (1926), demonstrated that two knot diagrams belonging to the same knot, up to planar [[Regular isotopy (nonfiction)|isotopy]], can be related by a sequence of the three Reidemeister moves.


Each move operates on a small region of the diagram and is one of three types:
Each move operates on a small region of the diagram and is one of three types:
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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.
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. 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.
One important context in which the Reidemeister moves appear is in defining [[Knot invariant (nonfiction)|knot invariants]]. 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 (nonfiction)|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.
The type I move is the only move that affects the [[Writhe (nonfiction)|writhe]] of the diagram. The type III move is the only one which does not change the crossing number of the diagram.


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.
In applications such as the [[Kirby calculus (nonfiction)|Kirby calculus]], in which the desired [[Equivalence class (nonfiction)|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.


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. Furthermore, combined work of Östlund (2001), Manturov (2004), and 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. 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.
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. Furthermore, combined work of Östlund (2001), Manturov (2004), and 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. 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.


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.
Coward & Lackenby (2014) proved the existence of an exponential tower [[Upper and lower bounds (nonfiction)|upper bound]] (depending on crossing number) on the number of Reidemeister moves required to pass between two diagrams of the same link.


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.  
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.  


Hayashi (2005) proved there is also an upper bound, depending on crossing number, on the number of Reidemeister moves required to split a link.
Hayashi (2005) proved there is also an upper bound, depending on crossing number, on the number of Reidemeister moves required to [[Split link (nonfiction)|split a link]].


== References ==
== References ==


Media related to Reidemeister moves at Wikimedia Commons
Media related to Reidemeister moves at Wikimedia Commons
Alexander, James W.; Briggs, Garland B. (1926), "On types of knotted curves", Annals of Mathematics, 28 (1/4): 562–586, doi:10.2307/1968399, JSTOR 1968399, MR 1502807
 
Coward, Alexander; Lackenby, Marc (2014), "An upper bound on Reidemeister moves", American Journal of Mathematics, 136 (4): 1023–1066, arXiv:1104.1882, doi:10.1353/ajm.2014.0027, MR 3245186, S2CID 55882290
* Alexander, James W.; Briggs, Garland B. (1926), "On types of knotted curves", Annals of Mathematics, 28 (1/4): 562–586, doi:10.2307/1968399, JSTOR 1968399, MR 1502807
Galatolo, Stefano (1999), "On a problem in effective knot theory", Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 9 (4): 299–306, MR 1722788
* Coward, Alexander; Lackenby, Marc (2014), "An upper bound on Reidemeister moves", American Journal of Mathematics, 136 (4): 1023–1066, arXiv:1104.1882, doi:10.1353/ajm.2014.0027, MR 3245186, S2CID 55882290
Hagge, Tobias (2006), "Every Reidemeister move is needed for each knot type", Proc. Amer. Math. Soc., 134 (1): 295–301, doi:10.1090/S0002-9939-05-07935-9, MR 2170571
* Galatolo, Stefano (1999), "On a problem in effective knot theory", Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 9 (4): 299–306, MR 1722788
Hass, Joel; Lagarias, Jeffrey C. (2001), "The number of Reidemeister moves needed for unknotting", Journal of the American Mathematical Society, 14 (2): 399–428, arXiv:math/9807012, doi:10.1090/S0894-0347-01-00358-7, MR 1815217, S2CID 15654705
* Hagge, Tobias (2006), "Every Reidemeister move is needed for each knot type", Proc. Amer. Math. Soc., 134 (1): 295–301, doi:10.1090/S0002-9939-05-07935-9, MR 2170571
Hayashi, Chuichiro (2005), "The number of Reidemeister moves for splitting a link", Mathematische Annalen, 332 (2): 239–252, doi:10.1007/s00208-004-0599-x, MR 2178061, S2CID 119728321
* Hass, Joel; Lagarias, Jeffrey C. (2001), "The number of Reidemeister moves needed for unknotting", Journal of the American Mathematical Society, 14 (2): 399–428, arXiv:math/9807012, doi:10.1090/S0894-0347-01-00358-7, MR 1815217, S2CID 15654705
Lackenby, Marc (2015), "A polynomial upper bound on Reidemeister moves", Annals of Mathematics, Second Series, 182 (2): 491–564, arXiv:1302.0180, doi:10.4007/annals.2015.182.2.3, MR 3418524, S2CID 119662237
* Hayashi, Chuichiro (2005), "The number of Reidemeister moves for splitting a link", Mathematische Annalen, 332 (2): 239–252, doi:10.1007/s00208-004-0599-x, MR 2178061, S2CID 119728321
Manturov, Vassily Olegovich (2004), Knot theory, Boca Raton, FL: Chapman & Hall/CRC, doi:10.1201/9780203402849, ISBN 0-415-31001-6, MR 2068425
* Lackenby, Marc (2015), "A polynomial upper bound on Reidemeister moves", Annals of Mathematics, Second Series, 182 (2): 491–564, arXiv:1302.0180, doi:10.4007/annals.2015.182.2.3, MR 3418524, S2CID 119662237
Östlund, Olof-Petter (2001), "Invariants of knot diagrams and relations among Reidemeister moves", J. Knot Theory Ramifications, 10 (8): 1215–1227, arXiv:math/0005108, doi:10.1142/S0218216501001402, MR 1871226, S2CID 119177881
* Manturov, Vassily Olegovich (2004), Knot theory, Boca Raton, FL: Chapman & Hall/CRC, doi:10.1201/9780203402849, ISBN 0-415-31001-6, MR 2068425
Reidemeister, Kurt (1927), "Elementare Begründung der Knotentheorie", Abh. Math. Sem. Univ. Hamburg, 5 (1): 24–32, doi:10.1007/BF02952507, MR 3069462, S2CID 120149796
* Östlund, Olof-Petter (2001), "Invariants of knot diagrams and relations among Reidemeister moves", J. Knot Theory Ramifications, 10 (8): 1215–1227, arXiv:math/0005108, doi:10.1142/S0218216501001402, MR 1871226, S2CID 119177881
Trace, Bruce (1983), "On the Reidemeister moves of a classical knot", Proceedings of the American Mathematical Society, 89 (4): 722–724, doi:10.2307/2044613, JSTOR 2044613, MR 0719004
* Reidemeister, Kurt (1927), "Elementare Begründung der Knotentheorie", Abh. Math. Sem. Univ. Hamburg, 5 (1): 24–32, doi:10.1007/BF02952507, MR 3069462, S2CID 120149796
* Trace, Bruce (1983), "On the Reidemeister moves of a classical knot", Proceedings of the American Mathematical Society, 89 (4): 722–724, doi:10.2307/2044613, JSTOR 2044613, MR 0719004


== In the News ==
== In the News ==
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[[Category:Knot theory (nonfiction)]]
[[Category:Knot theory (nonfiction)]]
[[Category:Mathemetics (nonfiction)]]
[[Category:Mathematics (nonfiction)]]

Revision as of 09:02, 25 March 2024

Diagram of Reidemeister moves.

In the mathematical area of knot theory, a Reidemeister move is any of three local moves on a link diagram. Kurt Reidemeister (1927) and, independently, James Waddell Alexander and Garland Baird Briggs (1926), demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves.

Each move operates on a small region of the diagram and is one of three types:

  • 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. 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.

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.

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. Furthermore, combined work of Östlund (2001), Manturov (2004), and 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. 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.

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.

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.

Hayashi (2005) proved there is also an upper bound, depending on crossing number, on the number of Reidemeister moves required to split a link.

References

Media related to Reidemeister moves at Wikimedia Commons

  • Alexander, James W.; Briggs, Garland B. (1926), "On types of knotted curves", Annals of Mathematics, 28 (1/4): 562–586, doi:10.2307/1968399, JSTOR 1968399, MR 1502807
  • Coward, Alexander; Lackenby, Marc (2014), "An upper bound on Reidemeister moves", American Journal of Mathematics, 136 (4): 1023–1066, arXiv:1104.1882, doi:10.1353/ajm.2014.0027, MR 3245186, S2CID 55882290
  • Galatolo, Stefano (1999), "On a problem in effective knot theory", Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 9 (4): 299–306, MR 1722788
  • Hagge, Tobias (2006), "Every Reidemeister move is needed for each knot type", Proc. Amer. Math. Soc., 134 (1): 295–301, doi:10.1090/S0002-9939-05-07935-9, MR 2170571
  • Hass, Joel; Lagarias, Jeffrey C. (2001), "The number of Reidemeister moves needed for unknotting", Journal of the American Mathematical Society, 14 (2): 399–428, arXiv:math/9807012, doi:10.1090/S0894-0347-01-00358-7, MR 1815217, S2CID 15654705
  • Hayashi, Chuichiro (2005), "The number of Reidemeister moves for splitting a link", Mathematische Annalen, 332 (2): 239–252, doi:10.1007/s00208-004-0599-x, MR 2178061, S2CID 119728321
  • Lackenby, Marc (2015), "A polynomial upper bound on Reidemeister moves", Annals of Mathematics, Second Series, 182 (2): 491–564, arXiv:1302.0180, doi:10.4007/annals.2015.182.2.3, MR 3418524, S2CID 119662237
  • Manturov, Vassily Olegovich (2004), Knot theory, Boca Raton, FL: Chapman & Hall/CRC, doi:10.1201/9780203402849, ISBN 0-415-31001-6, MR 2068425
  • Östlund, Olof-Petter (2001), "Invariants of knot diagrams and relations among Reidemeister moves", J. Knot Theory Ramifications, 10 (8): 1215–1227, arXiv:math/0005108, doi:10.1142/S0218216501001402, MR 1871226, S2CID 119177881
  • Reidemeister, Kurt (1927), "Elementare Begründung der Knotentheorie", Abh. Math. Sem. Univ. Hamburg, 5 (1): 24–32, doi:10.1007/BF02952507, MR 3069462, S2CID 120149796
  • Trace, Bruce (1983), "On the Reidemeister moves of a classical knot", Proceedings of the American Mathematical Society, 89 (4): 722–724, doi:10.2307/2044613, JSTOR 2044613, MR 0719004

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