Volume Fibrations on the SL(2,C) Character Variety of Hyperbolic Knot Complements

We introduce the volume functional h_Vol(A) = Im CS_C(A) = Vol(M_A)/(4π²) as a candidate Morse function on the geometric component of the SL(2,C) character variety of hyperbolic knot complements. The construction identifies the A-polynomial curve as a spectral-curve-type object carrying a canonical Liouville form v du, structurally parallel to the Hitchin spectral data, without requiring a full integrable system — a feature which is structurally precluded by the one-complex-dimensional base. We introduce a computable algebraic proxy h_∂(ℓ,m) = m + m⁻¹ and derive the explicit fiber structure for the figure-eight knot 4₁: for generic values, fibers consist of four points labeled by meridional and longitudinal branch choices, with the unitary branches satisfying cos β = cos 4α − cos 2α − 1. The fiber over the critical value h_crit = 2 undergoes an A₁-type nodal degeneration at (m,ℓ) = (1,−1), where two longitudinal sheets coalesce. We compute the branch periods on the real ovals of the unitary slice and identify the semiclassical Chern-Simons action increment. We conjecture that the discriminant locus of the volume fibration constitutes the topological counterpart of the metric degeneracy surface ε = 0 arising in disformal scalar-tensor gravity. These results provide a geometric foundation for the topological selection mechanism in the STKWC programme. Derivations and computations were produced using a coordinated multi-AI research methodology ("Parliament of Dragons"), employing Claude Opus 4, ChatGPT 5.2, ChatGPT 5.4, Gemini 2.5 Pro, and Grok 3 in specialized roles. All results were independently verified by at least two systems. The author bears sole responsibility for correctness and interpretation. Seventh preprint in the STKWC series. Part of the Scalar-Tensor Knot-Web Cosmology (STKWC) research programme. Multi-AI methodology documented in Acknowledgments section.