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 ([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)