The integral LKB module as absolute second homology

Let $\tilde{C}\to C$ be the LKB cover of The Lawrence-Krammer-Bigelow cover, with basepoint lift ${\tilde{c}}_0$, deck group ${\mathbb{Z}}^2=⟨q⟩\oplus ⟨t⟩$ and coefficient ring $\mathrm{\Lambda }=\mathbb{Z}[q^{\pm 1},t^{\pm 1}]$.

The integral Lawrence-Krammer-Bigelow module is the ordinary absolute singular homology $H_2(\tilde{C};\mathbb{Z})$ with the $\mathrm{\Lambda }$-module structure in which the generators $q,t$ act by the deck translations of the cover. No relative group is substituted for it.

The groups $H_2(\tilde{C},\tilde{\nu })\text{and}{H}_2(\tilde{C},\mathrm{\partial }\tilde{C}\cup \tilde{\nu })$ introduced from the small end neighbourhoods ${\tilde{\nu }}_{\epsilon }$ are auxiliary pairing targets only: they are used as the second argument (and, for the primed pairing, as the first argument) of the LKB intersection pairing, but the representation and all matrix statements of this page are about the absolute module $H_2(\tilde{C};\mathbb{Z})$. In particular, no identification of $H_2(\tilde{C};\mathbb{Z})$ with a relative or quotient module is asserted by this definition.