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Difference between revisions of "Conway skein triple"

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Three oriented link diagrams, or tangle diagrams, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130240/c1302401.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130240/c1302402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130240/c1302403.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130240/c1302404.png" />, or more generally, in any [[Three-dimensional manifold|three-dimensional manifold]], that are the same outside a small ball and in the ball look like
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Three oriented link diagrams, or tangle diagrams, $L_+$, $L_-$, $L_0$ in $\mathbf R^3$, or more generally, in any [[Three-dimensional manifold|three-dimensional manifold]], that are the same outside a small ball and in the ball look like
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c130240a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c130240a.gif" />
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Figure: c130240a
 
Figure: c130240a
  
Similarly one can define the Kauffman bracket skein triple of non-oriented diagrams <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130240/c1302405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130240/c1302406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130240/c1302407.png" />, and the Kauffman skein quadruple, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130240/c1302408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130240/c1302409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130240/c13024010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130240/c13024011.png" />, used to define the [[Brandt–Lickorish–Millett–Ho polynomial|Brandt–Lickorish–Millett–Ho polynomial]] and the [[Kauffman polynomial|Kauffman polynomial]]:
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Similarly one can define the Kauffman bracket skein triple of non-oriented diagrams $L_+$, $L_0$ and $L_\infty$, and the Kauffman skein quadruple, $L_+$, $L_-$, $L_0$ and $L_\infty$, used to define the [[Brandt–Lickorish–Millett–Ho polynomial|Brandt–Lickorish–Millett–Ho polynomial]] and the [[Kauffman polynomial|Kauffman polynomial]]:
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c130240b.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c130240b.gif" />
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Figure: c130240b
 
Figure: c130240b
  
Generally, a skein set is composed of a finite number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130240/c13024013.png" />-tangles and can be used to build link invariants and skein modules (cf. also [[Skein module|Skein module]]).
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Generally, a skein set is composed of a finite number of $k$-tangles and can be used to build link invariants and skein modules (cf. also [[Skein module|Skein module]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.H. Conway,  "An enumeration of knots and links"  J. Leech (ed.) , ''Computational Problems in Abstract Algebra'' , Pergamon  (1969)  pp. 329–358</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.H. Conway,  "An enumeration of knots and links"  J. Leech (ed.) , ''Computational Problems in Abstract Algebra'' , Pergamon  (1969)  pp. 329–358</TD></TR></table>

Latest revision as of 20:32, 11 April 2014

Three oriented link diagrams, or tangle diagrams, $L_+$, $L_-$, $L_0$ in $\mathbf R^3$, or more generally, in any three-dimensional manifold, that are the same outside a small ball and in the ball look like

Figure: c130240a

Similarly one can define the Kauffman bracket skein triple of non-oriented diagrams $L_+$, $L_0$ and $L_\infty$, and the Kauffman skein quadruple, $L_+$, $L_-$, $L_0$ and $L_\infty$, used to define the Brandt–Lickorish–Millett–Ho polynomial and the Kauffman polynomial:

Figure: c130240b

Generally, a skein set is composed of a finite number of $k$-tangles and can be used to build link invariants and skein modules (cf. also Skein module).

References

[a1] J.H. Conway, "An enumeration of knots and links" J. Leech (ed.) , Computational Problems in Abstract Algebra , Pergamon (1969) pp. 329–358
How to Cite This Entry:
Conway skein triple. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conway_skein_triple&oldid=14693
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article