Difference between revisions of "Alexander-Conway polynomial"
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$$\Delta_{L_+}-\Delta_{L_-}=z\Delta_{L_0}$$ | $$\Delta_{L_+}-\Delta_{L_-}=z\Delta_{L_0}$$ | ||
| − | 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 $\pm t^i$). | + | 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==== | ====References==== | ||
<table> | <table> | ||
| − | <TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> J.W. Alexander, "Topological invariants of knots and links" ''Trans. Amer. Math. Soc.'' , '''30''' (1928) pp. 275–306 {{ZBL|54.0603.03}}</TD></TR> |
| − | <TR><TD valign="top">[a2]</TD> <TD valign="top"> | + | <TR><TD valign="top">[a2]</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 {{ZBL|0202.54703}}</TD></TR> |
| − | <TR><TD valign="top">[a3]</TD> <TD valign="top"> | + | <TR><TD valign="top">[a3]</TD> <TD valign="top"> L.H. Kauffman, "The Conway polynomial" ''Topology'' , '''20''' : 1 (1981) pp. 101–108 {{ZBL|0456.57004}}</TD></TR> |
</table> | </table> | ||
[[Category:Algebraic topology]] | [[Category:Algebraic topology]] | ||
Latest revision as of 16:58, 18 May 2024
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 Zbl 54.0603.03 |
| [a2] | J.H. Conway, "An enumeration of knots and links" J. Leech (ed.) , Computational problems in abstract algebra , Pergamon (1969) pp. 329–358 Zbl 0202.54703 |
| [a3] | L.H. Kauffman, "The Conway polynomial" Topology , 20 : 1 (1981) pp. 101–108 Zbl 0456.57004 |
How to Cite This Entry:
Alexander-Conway polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alexander-Conway_polynomial&oldid=33693
Alexander-Conway polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alexander-Conway_polynomial&oldid=33693
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article