Difference between revisions of "Homotopy polynomial"
From Encyclopedia of Mathematics
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| − | An invariant of oriented links (cf. also [[ | + | An invariant of oriented links (cf. also [[Link]]). |
| − | It is a polynomial of two variables associated to homotopy classes of links in | + | It is a polynomial of two variables associated to homotopy classes of links in $\mathbf{R}^3$, depending only on linking numbers between components ([[#References|[a1]]], cf. also [[Knot theory]]). It satisfies the skein relation (cf. also [[Conway skein triple]]) |
| + | $$ | ||
| + | q^{-1}H_{L_{+}} - q H_{L_{-}} = z H_{L_{0}} | ||
| + | $$ | ||
| + | for a mixed crossing. The homotopy polynomial of a link with diagram $D$ is closely related to the dichromatic polynomial of the graph associated to $D$ (cf. also [[Graph colouring]]). The homotopy polynomial can be generalized to homotopy skein modules of three-dimensional manifolds (cf. also [[Skein module]]). | ||
| − | <table | + | ====References==== |
| + | <table> | ||
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> J.H. Przytycki, "Homotopy and $q$-homotopy skein modules of $3$-manifolds: An example in Algebra Situs" , ''Proc. Conf. in Low-Dimensional Topology in Honor of Joan Birman's 70th Birthday (Columbia Univ./Barnard College, March, 14-15, 1998)'' , Internat. Press (2000)</TD></TR> | ||
| + | </table> | ||
| − | + | {{TEX|done}} | |
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Latest revision as of 21:20, 7 May 2016
An invariant of oriented links (cf. also Link).
It is a polynomial of two variables associated to homotopy classes of links in $\mathbf{R}^3$, depending only on linking numbers between components ([a1], cf. also Knot theory). It satisfies the skein relation (cf. also Conway skein triple) $$ q^{-1}H_{L_{+}} - q H_{L_{-}} = z H_{L_{0}} $$ for a mixed crossing. The homotopy polynomial of a link with diagram $D$ is closely related to the dichromatic polynomial of the graph associated to $D$ (cf. also Graph colouring). The homotopy polynomial can be generalized to homotopy skein modules of three-dimensional manifolds (cf. also Skein module).
References
| [a1] | J.H. Przytycki, "Homotopy and $q$-homotopy skein modules of $3$-manifolds: An example in Algebra Situs" , Proc. Conf. in Low-Dimensional Topology in Honor of Joan Birman's 70th Birthday (Columbia Univ./Barnard College, March, 14-15, 1998) , Internat. Press (2000) |
How to Cite This Entry:
Homotopy polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homotopy_polynomial&oldid=12086
Homotopy polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homotopy_polynomial&oldid=12086
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article