Adjoint-twisted homology and Culler-Shalen data

Assume full AC. Let $K\subset S^3$ be a smooth oriented knot, $E_K$ its exterior, and $\rho :{\pi }_1(E_K)\to S{L}_2(\mathbb{C})$ a representation. The adjoint local system $\mathrm{Ad}\rho$ has fibre ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ and left group-module action $\gamma \cdot v=\rho (\gamma )v\rho (\gamma {)}^{-1}$. In the first-loop-first intrinsic convention, this means forward path transport $T_{\gamma }(v)=\rho (\gamma {)}^{-1}v\rho (\gamma )$; equivalently the corresponding right action is $v\cdot \gamma =\rho (\gamma {)}^{-1}v\rho (\gamma )$. Its twisted homology $H_{\ast }(E_K;\mathrm{Ad}\rho )$ is the intrinsic local-system homology with these transports, equivalently the homology of the cellular local-coefficient complex for a supplied CW structure; the same convention defines boundary-torus twisted homology and the inclusion map $i_{\ast }:H_1(\mathrm{\partial }{E}_K;\mathrm{Ad}\rho )\to H_1(E_K;\mathrm{Ad}\rho )$. An admissible adjoint representation is one for which there is a nonzero vector $v_{\rho }\in {\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ fixed by the peripheral adjoint action such that $\{[v_{\rho }\otimes \lambda ],[v_{\rho }\otimes \mu ]\}$ is a basis of $H_1(\mathrm{\partial }{E}_K;\mathrm{Ad}\rho )$ and $\ker i_{\ast }$ is one-dimensional. For an irreducible affine character curve $X$, its function field $\mathbb{C}(X)$ is the fraction field of its coordinate domain. A tautological representation over $\mathbb{C}(X)$ is a homomorphism ${\rho }_X:{\pi }_1(E_K)\to S{L}_2(\mathbb{C}(X))$ whose character is the generic character of $X$. A smooth projective completion consists of a smooth complete curve $\bar{X}$ with a dense open subset $U$ identified with the normalization $X^{\nu }$ of $X$, with its finite birational normalization map $\nu :U\to X$ recorded. When $X$ is smooth, $\nu$ identifies $U$ with $X$, recovering the usual completion. An ideal point is a point $x\in \bar{X}\setminus U$; points of $U$ over singular affine points of $X$ are affine branches and are retained inside $U$. Such an ideal point defines the discrete valuation $v_x(f)={\mathrm{ord}}_x(f)$ on $\mathbb{C}(X)$. The valuation ring is ${\mathcal{O}}_x=\{f:v_x(f)\ge 0\}$, and a uniformizer is an element $\pi \in {\mathcal{O}}_x$ with $v_x(\pi )=1$. Its residue field is $k_x={\mathcal{O}}_x\mathrm{/}(\pi )$. An ${\mathcal{O}}_x$-lattice in $\mathbb{C}(X{)}^2$ is a free rank-two ${\mathcal{O}}_x$-submodule spanning $\mathbb{C}(X{)}^2$; two lattices are homothetic if one is a nonzero scalar multiple of the other. The Bruhat–Tits tree has homothety classes of these lattices as vertices, with two vertices adjacent if representatives satisfy $\pi \mathrm{\Lambda }⊊{\mathrm{\Lambda }}^{\mathrm{\prime }}⊊\mathrm{\Lambda }$ and $\mathrm{\Lambda }\mathrm{/}{\mathrm{\Lambda }}^{\mathrm{\prime }}$ is one-dimensional over $k_x$. Whenever the tautological representation exists, it acts on this tree. The ideal point, valuation, action, and any essential surface produced from the action are its Culler–Shalen data; that surface is called the associated Culler–Shalen surface.

Intrinsic local-system chains and correctly typed cover-chain conventions

Cellular comparison for local coefficients

Normalization of an irreducible affine curve

Finite normalization and its isomorphism on the regular locus

Twisted cellular chain complex

Fox boundary matrix convention

Bénard, Florens, and Rodau, §§3.1 and 3.5

Culler and Shalen, §3