Alexander theorem (knot theory)

The set of Alexander is a theorem from the mathematical field of knot theory . It says that every twist is the end of a braid . It enables groups of braids to be used for the examination of knots and tangles. The set of Markov are necessary and sufficient conditions when the financial statements of two braids give equivalent tangles. It is named after James Alexander .

The 5-strand braid .

{\ displaystyle \ sigma _ {2} ^ {- 1} \ sigma _ {4} ^ {- 1} \ sigma _ {3} \ sigma _ {2} ^ {2} \ sigma _ {4} ^ {- 3} \ sigma _ {1} ^ {2} \ sigma _ {3}}

Finishing a braid

As in the picture on the right, a braid with strands is created by executing any sequence of interlacing and their inverses one behind the other. See the article Braid Group . ${\ displaystyle n}$ ${\ displaystyle \ sigma _ {1}, \ ldots, \ sigma _ {n-1}}$

The end of a braid is formed by connecting the first point of the lower edge with the first point of the upper edge, the second point of the lower edge with the second point of the upper edge, ..., the nth point of the lower edge with the n- th point of the upper edge is connected by untied arcs in the . ${\ displaystyle \ mathbb {R} ^ {3}}$

sentence

Alexander's theorem : Every loop can be constructed as the end of a braid.

Alexander's theorem follows from the fact first proved by Brunn that every node has a projection with only one multiple point. An algorithm going back to Pierre Vogel enables the implementation on the computer.

Examples

The Hopf loop is the end of the braid . ${\ displaystyle \ sigma _ {1} ^ {2}}$
The right-handed shamrock loop is the end of the braid . ${\ displaystyle \ sigma _ {1} ^ {3}}$
The figure eight is the end of the braid . ${\ displaystyle \ sigma _ {1} \ sigma _ {2} ^ {- 1} \ sigma _ {1} \ sigma _ {2} ^ {- 1}}$
The whitehead loop is the end of the braid . ${\ displaystyle \ sigma _ {1} \ sigma _ {2} ^ {- 1} \ sigma _ {1} \ sigma _ {2} ^ {- 2}}$
The Borromean rings are the end of the braid . ${\ displaystyle \ sigma _ {1} ^ {- 1} \ sigma _ {2} \ sigma _ {1} ^ {- 1} \ sigma _ {2} \ sigma _ {1} ^ {- 1} \ sigma _ {2}}$

Braid index

The Zopfindex (ger .: braid index ) of entanglement is the smallest number of strands of a braid, as its conclusion one can construct the entanglement. It is a knot invariant and can be used, for example, to distinguish the left and right-handed clover leaf loop from one another.

literature

JW Alexander: A lemma on systems of knotted curves. Proc. Nat. Acad. Sci. USA 9 (1923) pp. 93-95.
HK Brunn: Over knotted curves. Verh. Math. Kongr. Zürich (1897) pp. 256–259.

Web links

Alexander Theorem on Braids (Encyclopedia of Mathematics)
Braid Word (Math World)

Individual evidence

^ Proposition 2.14 in: Burde, Zieschang: Knots. Second edition. de Gruyter Studies in Mathematics, 5. Walter de Gruyter & Co., Berlin, 2003. ISBN 3-11-017005-1

[1] Proposition 2.14 in: Burde, Zieschang: Knots. Second edition. de Gruyter Studies in Mathematics, 5. Walter de Gruyter & Co., Berlin, 2003. ISBN 3-11-017005-1