In the mathematical area of knot theory, a Reidemeister move refers to one of three local moves on a link diagram. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, knot theory is the area of Topology that studies mathematical knots While inspired by knots which appear in daily life in shoelaces In 1927, J.W. Alexander and G. James Waddell Alexander II ( September 19, 1888 – September 23, 1971) was an important Topologist of the pre-WWII era and part of B. Briggs, and independently Kurt Reidemeister, 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. Kurt Werner Friedrich Reidemeister ( October 13, 1893 - July 8, 1971) was a Mathematician born in Braunschweig (Brunswick In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical
 |
 |
| Type I | Type II |
 |
|
| Type III | |
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.
Note that no other part of the diagram should be in the picture of a move, and that 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. In the mathematical field of Knot theory, a knot invariant is a quantity (in a broad sense defined for each knot which is the same for equivalent knots 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. In the mathematical field of Knot theory, the Jones polynomial is a Knot polynomial discovered by Vaughan Jones in 1983
The type I move is the only move that affects the writhe of the link. In Knot theory, the writhe is a property of an oriented link diagram The type III move is the only one which does not change the crossing number of the diagram.
Bruce Trace showed that two knot diagrams are related by using only type II and III moves if and only if they have the same writhe and winding number. In Knot theory, the writhe is a property of an oriented link diagram Furthermore, combined work of O. Östlund, V. O. Manturov, and T. Hagge 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 3 moves increase crossing number while those after decrease crossing number.
In another vein, Stefano Galatolo, and independently Joel Hass and Jeffrey Lagarias (with a better bound), have shown that there is an upper bound (depending on crossing number) on the number of Reidemeister moves required to change a diagram of the unknot to the standard unknot. Jeffrey Clark Lagarias (b November 1949 in Pittsburgh Pennsylvania) is a mathematics professor at the University of Michigan. This gives an inefficient algorithm to solve the unknotting problem. In Mathematics, the unknotting problem is the problem of algorithmically recognizing the Unknot, given some input e
Chuichiro Hayashi proved there is also an upper bound, depending on crossing number, on the number of Reidemeister moves required to split a link. In the mathematical field of Knot theory, a split link is a link that has a (topological 2-sphere in its complement separating one or more link components
References
- J. W. Alexander; G. B. Briggs, On types of knotted curves. Ann. of Math. (2) 28 (1926/27), no. The Annals of Mathematics (ISSN 0003-486X abbreviated as Ann of Math 1-4, 562--586.
- Kurt Reidemeister, Elementare Begründung der Knotentheorie, Abh. Math. Sem. Univ. Hamburg 5 (1926), 24-32
- Bruce Trace, On the Reidemeister moves of a classical knot. Proc. Amer. Math. Soc. 89 (1983), no. 4, 722--724.
- Tobias Hagge, Every Reidemeister move is needed for each knot type. Proc. Amer. Math. Soc. 134 (2006), no. 1, 295--301.
- Stefano Galatolo, On a problem in effective knot theory. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 9 (1998), no. 4, 299--306 (1999).
- Joel Hass; Jeffrey Lagarias, The number of Reidemeister moves needed for unknotting. J. Amer. Math. Soc. 14 (2001), no. The Journal of the American Mathematical Society, often referred to by its acronym JAMS, is a mathematics journal published quarterly by the American Mathematical Society 2, 399--428
- Chuichiro Hayashi, The number of Reidemeister moves for splitting a link. Math. Ann. 332 (2005), no. 2, 239--252.
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic