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A one-dimensional manifold properly embedded in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130030/t1300301.png" />-ball, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130030/t1300302.png" />.
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A one-dimensional manifold properly embedded in a $3$-ball, $D^3$.
  
Two tangles are considered equivalent if they are ambient isotopic with their boundary fixed. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130030/t1300304.png" />-tangle has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130030/t1300305.png" /> points on the boundary; a link is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130030/t1300306.png" />-tangle. The term arcbody is used for a one-dimensional manifold properly embedded in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130030/t1300307.png" />-dimensional manifold.
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Two tangles are considered equivalent if they are ambient isotopic with their boundary fixed. An $n$-tangle has $2n$ points on the boundary; a link is a $0$-tangle. The term "arcbody" is used for a one-dimensional manifold properly embedded in a $3$-dimensional manifold.
  
Tangles can be represented by their diagrams, i.e. regular projections into a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130030/t1300308.png" />-dimensional disc with additional over- and under-information at crossings. Two tangle diagrams represent equivalent tangles if they are related by Reidemeister moves (cf. [[Reidemeister theorem|Reidemeister theorem]]). The word  "tangle"  is often used to mean a tangle diagram or part of a link diagram.
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Tangles can be represented by their diagrams, i.e. regular projections into a $2$-dimensional disc with additional over- and under-information at crossings. Two tangle diagrams represent equivalent tangles if they are related by Reidemeister moves (cf. [[Reidemeister theorem]]). The word  "tangle"  is often used to mean a tangle diagram or part of a link diagram (cf. [[Knot and link diagrams]].
  
The set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130030/t1300309.png" />-tangles forms a [[Monoid|monoid]]; the identity tangle and composition of tangles is illustrated in Fig.a1.
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The set of $n$-tangles forms a [[monoid]]; the identity tangle and composition of tangles is illustrated in Fig.a1.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t130030a.gif" />
 
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Figure: t130030a
 
Figure: t130030a
  
Several special families of tangles have been considered, including the [[Rational tangles|rational tangles]], the [[Algebraic tangles|algebraic tangles]] and the periodic tangles (see [[Rotor|Rotor]]). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130030/t13003010.png" />-braid group is a subgroup of the monoid of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130030/t13003011.png" />-tangles (cf. also [[Braided group|Braided group]]). One has also considered framed tangles and graph tangles. The category of tangles, with boundary points as objects and tangles as morphisms, is important in developing quantum invariants of links and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t130/t130030/t13003012.png" />-manifolds (e.g. Reshetikhin–Turaev invariants). Tangles are also used to construct topological quantum field theories.
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Several special families of tangles have been considered, including the [[rational tangles]], the [[algebraic tangles]] and the periodic tangles (see [[Rotor]]). The $n$-braid group is a subgroup of the monoid of $n$-tangles (cf. also [[Braid theory]], [[Braided group]]). One has also considered framed tangles and graph tangles. The category of tangles, with boundary points as objects and tangles as morphisms, is important in developing quantum invariants of links and $3$-manifolds (e.g. Reshetikhin–Turaev invariants). Tangles are also used to construct topological quantum field theories.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Bonahon,  L. Siebenmann,  "Geometric splittings of classical knots and the algebraic knots of Conway" , ''Lecture Notes'' , '''75''' , London Math. Soc.  (to appear)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.H. Conway,  "An enumeration of knots and links"  J. Leech (ed.) , ''Computational Problems in Abstract Algebra'' , Pergamon Press  (1969)  pp. 329–358</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Lozano,  "Arcbodies"  ''Math. Proc. Cambridge Philos. Soc.'' , '''94'''  (1983)  pp. 253–260</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Bonahon,  L. Siebenmann,  "Geometric splittings of classical knots and the algebraic knots of Conway" , ''Lecture Notes'' , '''75''' , London Math. Soc.  (to appear)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.H. Conway,  "An enumeration of knots and links"  J. Leech (ed.) , ''Computational Problems in Abstract Algebra'' , Pergamon Press  (1969)  pp. 329–358</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Lozano,  "Arcbodies"  ''Math. Proc. Cambridge Philos. Soc.'' , '''94'''  (1983)  pp. 253–260</TD></TR>
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</table>
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Revision as of 17:42, 21 January 2017

relative link

A one-dimensional manifold properly embedded in a $3$-ball, $D^3$.

Two tangles are considered equivalent if they are ambient isotopic with their boundary fixed. An $n$-tangle has $2n$ points on the boundary; a link is a $0$-tangle. The term "arcbody" is used for a one-dimensional manifold properly embedded in a $3$-dimensional manifold.

Tangles can be represented by their diagrams, i.e. regular projections into a $2$-dimensional disc with additional over- and under-information at crossings. Two tangle diagrams represent equivalent tangles if they are related by Reidemeister moves (cf. Reidemeister theorem). The word "tangle" is often used to mean a tangle diagram or part of a link diagram (cf. Knot and link diagrams.

The set of $n$-tangles forms a monoid; the identity tangle and composition of tangles is illustrated in Fig.a1.

Figure: t130030a

Several special families of tangles have been considered, including the rational tangles, the algebraic tangles and the periodic tangles (see Rotor). The $n$-braid group is a subgroup of the monoid of $n$-tangles (cf. also Braid theory, Braided group). One has also considered framed tangles and graph tangles. The category of tangles, with boundary points as objects and tangles as morphisms, is important in developing quantum invariants of links and $3$-manifolds (e.g. Reshetikhin–Turaev invariants). Tangles are also used to construct topological quantum field theories.

References

[a1] F. Bonahon, L. Siebenmann, "Geometric splittings of classical knots and the algebraic knots of Conway" , Lecture Notes , 75 , London Math. Soc. (to appear)
[a2] J.H. Conway, "An enumeration of knots and links" J. Leech (ed.) , Computational Problems in Abstract Algebra , Pergamon Press (1969) pp. 329–358
[a3] M. Lozano, "Arcbodies" Math. Proc. Cambridge Philos. Soc. , 94 (1983) pp. 253–260
How to Cite This Entry:
Tangle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangle&oldid=15693
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article