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Alexander-Conway polynomial

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The normalized version of the Alexander polynomial (cf. also Alexander invariants). It satisfies the Conway skein relation (cf. also Conway skein triple)

$$\Delta_{L_+}-\Delta_{L_-}=z\Delta_{L_0}$$

and the initial condition $\Delta_{T_1}=1$, where $T_1$ is the trivial knot (cf. also Knot theory). For $z=\sqrt t-1/\sqrt t$ one gets the original Alexander polynomial (defined only up to $\pm t^i$).

References

[a1] J.W. Alexander, "Topological invariants of knots and links" Trans. Amer. Math. Soc. , 30 (1928) pp. 275–306
[a2] J.H. Conway, "An enumeration of knots and links" J. Leech (ed.) , Computational problems in abstract algebra , Pergamon (1969) pp. 329–358
[a3] L.H. Kauffman, "The Conway polynomial" Topology , 20 : 1 (1981) pp. 101–108
How to Cite This Entry:
Alexander–Conway polynomial. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Alexander%E2%80%93Conway_polynomial&oldid=22014