Difference between revisions of "Conway skein equivalence"
From Encyclopedia of Mathematics
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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>< | + | <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
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