Character-variety logarithmic slope

Assume full AC. Let $K\subset S^3$ be a smooth oriented knot, let $\rho :{\pi }_1(E_K)\to S{L}_2(\mathbb{C})$ have admissible adjoint local system, and let $v_{\rho }$ be a nonzero peripheral-adjoint-invariant vector for which $[v_{\rho }\otimes \lambda ],[v_{\rho }\otimes \mu ]$ form the specified ordered basis of $H_1(\mathrm{\partial }{E}_K;\mathrm{Ad}\rho )$. If a generator of the one-dimensional kernel of inclusion is $a[v_{\rho }\otimes \lambda ]+b[v_{\rho }\otimes \mu ]$, the Bénard–Florens–Rodau slope is $s_K(\rho )=-b\mathrm{/}a\in \mathbb{C}\cup \{\mathrm{\infty }\}$, with $a=0$ interpreted as $\mathrm{\infty }$. Rescaling $v_{\rho }$ or the kernel generator does not change this projective ratio. A rational function on an irreducible character component means an element of its function field, the fraction field of its coordinate domain. A projective-valued rational function means a point of the projective line over that field; it may be identically $\mathrm{\infty }$. The slope's rational extension has this projective meaning. Unless its generic value is $\mathrm{\infty }$, it is an ordinary rational function, whose individual infinite values are poles. Its rational extension is not a value assigned to every singular or inadmissible character.

Admissible adjoint homology and peripheral basis

Bénard, Florens, and Rodau, §3.1

Tillmann, §§4.5–4.8