The Lawrence-Krammer-Bigelow representation

Assume AC. Let $D$ be the closed unit disk, $P=\{p_1,\dots ,p_n\}\subset \mathrm{int}D$ the puncture set in the standard configuration, $C$ the unordered two-point configuration space of $D\setminus P$ with its basepoint $c_0$, and $\tilde{C}\to C$ the LKB 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 two-point configuration space of a punctured disk, The Lawrence-Krammer-Bigelow cover).

By Braid group as boundary-fixed punctured-disk mapping classes the classical braid group $B_n$ is identified with the boundary-fixed mapping class group $\mathrm{Mod}(D,P;\mathrm{\partial }D)$: homeomorphisms of $D$ fixing $\mathrm{\partial }D$ pointwise and preserving $P$ setwise, modulo isotopy relative to $\mathrm{\partial }D\cup P$. Let $[\sigma ]\in B_n$ and let $\sigma$ be a representative. By Braids lift to the LKB cover and act Lambda-linearly there is a unique lift $\tilde{\sigma }$ of $\sigma$ to $\tilde{C}$ fixing ${\tilde{c}}_0$; it commutes with every deck transformation, and its induced automorphism ${\tilde{\sigma }}_{\ast }:H_2(\tilde{C};\mathbb{Z})⟶H_2(\tilde{C};\mathbb{Z})$ is $\mathrm{\Lambda }$-linear and invertible. The assignment $[\sigma ]\mapsto {\tilde{\sigma }}_{\ast }$ is independent of the representative and multiplicative, so it defines a homomorphism $B_n⟶{\mathrm{Aut}}_{\mathrm{\Lambda }}{H}_2(\tilde{C};\mathbb{Z})\mathrm{.}$

The Lawrence-Krammer-Bigelow representation is this homomorphism composed with the matrix presentation of the target: fix once and for all the $\mathrm{\Lambda }$-basis $\{v_{i,j}:1\le i<j\le n\}$ of $H_2(\tilde{C};\mathbb{Z})$ supplied for $n\ge 2$ by The integral LKB module is free of rank n choose two. For $n=1$ the basis is empty and $H_2=0$: the absolute cellular rank and localization injection of The absolute LKB cellular boundary and fraction-field rank give rank0 and therefore zero homology. They apply to the disk as well as the plane: choose a radial collar outside all punctures and compress its boundary strictly inward by a strictly increasing radius map fixed below the collar. Interpolating that map with the identity keeps every two-point configuration collision-free and gives inverse homotopies for interior inclusion. The homotopy preserves puncture and mutual winding characters; the moving basepoint is transported along its specified track. The open disk is orientation-preservingly radially homeomorphic to the plane, taking real punctures to real punctures in the same order. This transfers the absolute cover calculation. Define ${\rho }_{\mathrm{L}\mathrm{K}\mathrm{B}}:B_n⟶{\mathrm{G}\mathrm{L}}_{\left(\genfrac{}{}{0px}{}{n}{2}\right)}(\mathrm{\Lambda }),{\rho }_{\mathrm{L}\mathrm{K}\mathrm{B}}([\sigma ])=\text{the matrix of }{\tilde{\sigma }}_{\ast }\text{ in the basis }\{v_{i,j}\}\mathrm{.}$ Thus ${\rho }_{\mathrm{L}\mathrm{K}\mathrm{B}}$ is the action of $B_n$ on the absolute integral module $H_2(\tilde{C};\mathbb{Z})$; the relative modules and the pairing are not used in its definition, and the Axiom of Choice is used exactly in the identification of $B_n$ with the boundary-fixed mapping class group, which supplies the normalized lifts above.

Two caveats are part of the definition. First, the target is the integral matrix group over $\mathrm{\Lambda }$; by The integral LKB module is free of rank n choose two the basis $\{v_{i,j}\}$ is not related to Krammer's matrix basis by a $B_n$-equivariant $\mathrm{\Lambda }$-isomorphism when $n\ge 3$, and only the fraction-field models are identified. Second, the normalization by the lift fixing ${\tilde{c}}_0$ is essential: replacing it by another lift multiplies ${\tilde{\sigma }}_{\ast }$ by a deck transformation, i.e. by a monomial $q^a{t}^b$, so the matrix of ${\rho }_{\mathrm{L}\mathrm{K}\mathrm{B}}([\sigma ])$ below is the one computed from this fixed normalization.