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An equivalence relation on the set of links in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130230/c1302301.png" /> (cf. also [[Link|Link]]). It is the smallest equivalence relation on ambient isotopy classes of oriented links, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130230/c1302302.png" />, that satisfies the following condition: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130230/c1302303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130230/c1302304.png" /> are Conway skein triples (cf. also [[Conway skein triple|Conway skein triple]]) such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130230/c1302305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130230/c1302306.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130230/c1302307.png" />, and, furthermore, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130230/c1302308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130230/c1302309.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130230/c13023010.png" />.
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An equivalence relation on the set of links in $\mathbf{R} ^ { 3 }$ (cf. also [[Link|Link]]). It is the smallest equivalence relation on ambient isotopy classes of oriented links, denoted by $\sim _ { c }$, that satisfies the following condition: If $( L _ { + } , L _ { - } , L _ { 0 } )$ and $( L _ { + } ^ { \prime } , L ^ { \prime }_{ -} , L _ { 0 } ^ { \prime } )$ are Conway skein triples (cf. also [[Conway skein triple|Conway skein triple]]) such that if $L _ { - } \sim _ { c } L _ { - } ^ { \prime }$ and $L _ { 0 } \sim _ { c } L _ { 0 } ^ { \prime }$ then $L _ { + } \sim _ { c } L _ { + } ^ { \prime }$, and, furthermore, if $L _ { + } \sim _ { c } L _ { + } ^ { \prime }$ and $L _ { 0 } \sim _ { c } L _ { 0 } ^ { \prime }$ then $L _ { - } \sim _ { c } L _ { - } ^ { \prime }$.
  
 
Skein equivalent links have the same Jones–Conway polynomials (cf. also [[Jones–Conway polynomial|Jones–Conway polynomial]]) and the same Murasugi signatures (for links with non-zero determinant, cf. also [[Signature|Signature]]). The last property generalizes to Tristram–Levine signatures.
 
Skein equivalent links have the same Jones–Conway polynomials (cf. also [[Jones–Conway polynomial|Jones–Conway polynomial]]) and the same Murasugi signatures (for links with non-zero determinant, cf. also [[Signature|Signature]]). The last property generalizes to Tristram–Levine signatures.
  
 
====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><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.A. Giller,  "A family of links and the Conway calculus"  ''Trans. Amer. Math. Soc.'' , '''270''' :  1  (1982)  pp. 75–109</TD></TR></table>
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<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><tr><td valign="top">[a2]</td> <td valign="top">  C.A. Giller,  "A family of links and the Conway calculus"  ''Trans. Amer. Math. Soc.'' , '''270''' :  1  (1982)  pp. 75–109</td></tr></table>

Latest revision as of 16:56, 1 July 2020

An equivalence relation on the set of links in $\mathbf{R} ^ { 3 }$ (cf. also Link). It is the smallest equivalence relation on ambient isotopy classes of oriented links, denoted by $\sim _ { c }$, that satisfies the following condition: If $( L _ { + } , L _ { - } , L _ { 0 } )$ and $( L _ { + } ^ { \prime } , L ^ { \prime }_{ -} , L _ { 0 } ^ { \prime } )$ are Conway skein triples (cf. also Conway skein triple) such that if $L _ { - } \sim _ { c } L _ { - } ^ { \prime }$ and $L _ { 0 } \sim _ { c } L _ { 0 } ^ { \prime }$ then $L _ { + } \sim _ { c } L _ { + } ^ { \prime }$, and, furthermore, if $L _ { + } \sim _ { c } L _ { + } ^ { \prime }$ and $L _ { 0 } \sim _ { c } L _ { 0 } ^ { \prime }$ then $L _ { - } \sim _ { c } L _ { - } ^ { \prime }$.

Skein equivalent links have the same Jones–Conway polynomials (cf. also Jones–Conway polynomial) and the same Murasugi signatures (for links with non-zero determinant, cf. also Signature). The last property generalizes to Tristram–Levine signatures.

References

[a1] J.H. Conway, "An enumeration of knots and links" J. Leech (ed.) , Computational Problems in Abstract Algebra , Pergamon (1969) pp. 329–358
[a2] C.A. Giller, "A family of links and the Conway calculus" Trans. Amer. Math. Soc. , 270 : 1 (1982) pp. 75–109
How to Cite This Entry:
Conway skein equivalence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conway_skein_equivalence&oldid=50173
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article