Difference between revisions of "Conway skein triple"
From Encyclopedia of Mathematics
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| − | Three oriented link diagrams, or tangle diagrams, | + | {{TEX|done}} |
| + | 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 | + | 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 | + | 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> | ||
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
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