Let $C_n$ denote the configuration space of $n$ distinct points in hyperbolic $3$-space $\mathcal{H}$. If $\mathbf{x} := (\mathbf{x}_1, \dots, \mathbf{x}_n) \in C_n$, where $\mathbf{x}_i \in \mathcal{H}$ for $i = 1, \ldots, n$, M.F. Atiyah associated to $\mathbf{x}$ $n$ polynomials. More specifically, for $i \in [n] := \{1, \dots, n\}$, imagine one's self sitting at $\mathbf{x}_i$ and then draw hyperbolic rays from $\mathbf{x}_i$ to the other points in $\mathbf{x}$. One then gets, as limiting points of these hyperbolic rays, points $\zeta_{ij}$, for $j \in [n] \setminus \{i\}$. We think of the sphere at infinity as the Riemann sphere. Hence, each $\zeta_{ij}$ becomes identified with a complex number, or possibly $\infty$. We let $p_i(\zeta)$ be a nonzero complex polynomial of degree $n - 1$, with roots $$ \{ \zeta_{ij} ; \, j \in [n] \setminus \{i\} \}, $$ taking multiplicity into account (so that the latter is a multiset or, equivalently, an effective divisor on $\mathbb{P}^1$ of degree $n - 1$).
Atiyah conjectured that the $n$ polynomials $p_1(\zeta), \dots, p_n(\zeta)$ are linearly independent over $\mathbb{C}$, no matter what configuration $\mathbf{x} \in C_n$ one starts with.
There is a natural pairing on polynomials of degree (at most) $d$. Let $$ p(\zeta) = a_0 + a_1 \binom{d}{1} \zeta + \dots + a_i \binom{d}{i} \zeta^i + \dots + a_d \zeta^d $$ and $$ q(\zeta) = b_0 + b_1 \binom{d}{1} \zeta + \dots + b_i \binom{d}{i} \zeta^i + \dots + b_d \zeta^d. $$ One then defines $$ (p, q)_d = \sum_{i = 0}^d (-1)^i \binom{d}{i} a_i b_{d-i}.$$ One says that $p(\zeta)$ and $q(\zeta)$ are $d$-apolar if $(p, q)_d = 0$.
If we assume Atiyah's linear independence conjecture (which at the time of writing is known to be true for $n = 3$ thanks to work by Atiyah, and some work has been done for $n = 4$), it then follows that $(p_1(\zeta), \dots, p_n(\zeta))$ is a basis of the space of complex polynomials $\mathcal{P}_{n-1}$ of degree at most $n - 1$. The pairing $(-, -)_{n-1}$ is non-degenerate, hence there exist a dual basis $q_1(\zeta), \dots, q_n(\zeta)$ of $\mathcal{P}_{n-1}$ with respect to this pairing, i.e. $$ (p_i, q_j)_{n-1} = \delta_{ij}, \qquad \text{for $i, j \in [n]$.}$$
Question: does the above duality respect the relations between the $\zeta_{ij}$ that ensure that the $(\zeta_{ij})$ are geometric, i.e. that they arise from a hyperbolic configuration, say $\tau(\mathbf{x}) \in C_n$? In other words, are the $q_i(\zeta)$ the Atiyah polynomials of some hyperbolic configuration in $C_n$? If so, then I would like to think of it as a kind of dual of $\mathbf{x}$. Let us just call it the $\tau$-dual of $\mathbf{x}$, for now. If it is true, I may think of some better name later on.
