Conway skein equivalence

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