In a 1984 paper, Squier wrote down an explicit Hermitian form $J$ relative to which the reduced Burau representation is unitary. The form is valued in the ring $\mathbb{Z}[s^{\pm 1}]$, where $s^2 = t$ and $t$ is the standard variable appearing in the Burau representation. In this context, the involution on $\mathbb{Z}[s^{\pm 1}]$ is induced from the map $s \mapsto s^{-1}$. Later it was understood that the reduced Burau representation is also *skew-Hermitian* with respect to a sort of symmetrized intersection form on the Burau cover of the disk, valued in $\mathbb{Z}[t^{\pm}]$.

I am wondering if there is an extension of either of these forms to the unreduced Burau representation. In light of the fact that the unreduced Burau rep is given by action on a *relative* homology group, for which there doesn't seem to be a good notion of an intersection pairing, I'm not optimistic that the unreduced Burau rep can be made to be skew-Hermitian.

On the other hand, I don't know if it's possible to define something like Squier's form for unreduced Burau. Squier pulls his form out of thin air, and I haven't come across any discussion in the literature of a topological definition of Squier's form (I've looked in the papers of Gambaudo-Ghys, McMullen, and Venkataramana that study the Burau representation at roots of unity; in each of these the intersection form plays a crucial role, but none seem to explain where Squier's version of the form comes from).

In summary,

Is there a Hermitian form on the domain of the unreduced Burau representation for which Burau is unitary? Can one extend Squier's form in some way?

Is there a topological construction of Squier's intersection form? How did he write his form down?