Tangle theory | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-30

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Tangle_(mathematics)


00:01:46 1 Tangle diagrams
00:02:30 2 Rational and algebraic tangles
00:05:48 2.1 Operations on tangles
00:06:37 3 Conway notation
00:06:58 4 Applications
00:07:19 5 See also



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SUMMARY
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In mathematics, a tangle is generally one of two related concepts:

In John Conway's definition, an n-tangle is a proper embedding of the disjoint union of n arcs into a 3-ball; the embedding must send the endpoints of the arcs to 2n marked points on the ball's boundary.
In link theory, a tangle is an embedding of n arcs and m circles into 
  
    
      
        
          
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        0
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    {\displaystyle \mathbf {R} ^{2}\times [0,1]}
   – the difference from the previous definition is that it includes circles as well as arcs, and partitions the boundary into two (isomorphic) pieces, which is algebraically more convenient – it allows one to add tangles by stacking them, for instance.(A quite different use of 'tangle' appears in Graph minors X. Obstructions to tree-decomposition by N. Robertson and P. D. Seymour, Journal of Combinatorial Theory B 59 (1991) 153–190, who used it to describe separation in graphs. This usage has been extended to matroids.)
The balance of this article discusses Conway's sense of tangles; for the link theory sense, see that article.
Two n-tangles are considered equivalent if there is an ambient isotopy of one tangle to the other keeping the boundary of the 3-ball fixed. Tangle theory can be considered analogous to knot theory except instead of closed loops we use strings whose ends are nailed down. See also braid theory.