View source for Homotopy polynomial

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 ([[#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]]).

====References====
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<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>

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