Difference between revisions of "Alexander theorem on braids"
From Encyclopedia of Mathematics
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Every link has a closed braid presentation (cf. also [[Braid theory|Braid theory]]; [[Link|Link]]). | Every link has a closed braid presentation (cf. also [[Braid theory|Braid theory]]; [[Link|Link]]). | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.W. Alexander, "A lemma on systems of knotted curves" ''Proc. Nat. Acad. Sci. USA'' , '''9''' (1923) pp. 93–95</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H.K. Brunn, "Über verknotete Kurven" , ''Verh. Math. Kongr. Zürich'' (1897) pp. 256–259</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> J.W. Alexander, "A lemma on systems of knotted curves" ''Proc. Nat. Acad. Sci. USA'' , '''9''' (1923) pp. 93–95</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> H.K. Brunn, "Über verknotete Kurven" , ''Verh. Math. Kongr. Zürich'' (1897) pp. 256–259</TD></TR> | ||
| + | </table> | ||
Latest revision as of 08:32, 17 March 2023
2020 Mathematics Subject Classification: Primary: 57K [MSN][ZBL]
Every link has a closed braid presentation (cf. also Braid theory; Link).
This result, published by J.W. Alexander in 1923, allows one to study knots and links using the theory of braids, [a1] (cf. also Knot theory). Alexander's theorem has its roots in Brunn's result (1897) that every knot has a projection with only one multiple point (it is usually not a regular projection) [a2].
The smallest number of braid strings used in the presentation is called the braid index of the link.
References
| [a1] | J.W. Alexander, "A lemma on systems of knotted curves" Proc. Nat. Acad. Sci. USA , 9 (1923) pp. 93–95 |
| [a2] | H.K. Brunn, "Über verknotete Kurven" , Verh. Math. Kongr. Zürich (1897) pp. 256–259 |
How to Cite This Entry:
Alexander theorem on braids. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alexander_theorem_on_braids&oldid=52750
Alexander theorem on braids. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alexander_theorem_on_braids&oldid=52750
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article