Narrow Results

We give a new definition of the Jones polynomial by means of the intersection number of loaded (or twisted) cycles associated with a Selberg type integral. Our definition is naturally formulated in the framework of the twisted homology theory, which is developd by Aomoto to study the special functions of hypergeometric type. The naturality of the definition leads to evaluate the Jones polynomials in several cases: well-known results in the case of two-bridge link, a formula for (3, s)-torus and that for the Prezel with 3 parameters. Our definition is motivated by the work of Bigelow.

A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main motivating example, we calculate the rational homology groups of spaces of even and odd maps $S^m\to S^M$, $m<M$, or, which is the same, the stable homology groups of spaces of non-resultant homogeneous polynomial maps ${ℝ}^{m+1}\to {ℝ}^{M+1}$ of increasing degree. Also, we calculate the homology groups of spaces of ${\mathbb{Z}}_r$-equivariant maps of odd-dimensional spheres for any $r$. In auxiliary calculations, we find the homology groups of configuration spaces of projective and lens spaces with coefficients in several local systems.