An Image--Kernel--Reconstruction Program for Essential-Surface Complexes of Knot Exteriors

Abstract

We study the action of the meridian-preserving mapping class group of a knot exterior E on the disjointness complex of connected two-sided orientable essential surfaces, with natural type, peripheral, homological, and characteristic-submanifold decorations. The underlying complex is Schultens's initial surface complex S_0(E). We organize four rigidity questions: which automorphisms are geometric, which mapping classes are invisible, what characteristic and peripheral structure is intrinsically recoverable, and how much decoration is necessary. A conditional reduction principle separates recognition, global realization, and kernel determination; an intentionally over-marked hierarchy atlas gives a reconstruction benchmark. Classical three-manifold results yield model calculations. For a torus-knot exterior, the complex has two isolated vertices; a type or slope label removes its nongeometric transposition, while the labeled-action kernel is generated by strong inversion. For a connected sum of two nonfibered prime knots, the decomposing annulus is the unique essential annulus, and its twist translates Banks's integer winding coordinate, so the annular-twist subgroup acts faithfully on the Kakimizu complex. For a hyperbolic knot exterior, every kernel considered is finite and has no nontrivial twist subgroup. For the figure-eight knot, the complex has three isolated vertices of slopes 0 and +/-4; the full mapping class group is dihedral of order eight, the geometric image on the four decorated objects considered is Z/2, and the kernel is the orientation-preserving Klein four subgroup. The classifications and symmetry groups are classical; the contribution is the common image-kernel bookkeeping and reconstruction framework.

Disclosure

“orphism of essential-surface complexes, and determining how much of the complex is remembered by its slope shadow is a natural refinement question. Acknowledgements Use of generative AI. The author used ChatGPT (OpenAI) and Claude Fable 5 (Anthropic) as interactive aids during the preparation of this manuscript, including for mathematical discussion, consideration of alternative formulations and possible proof strategies, preliminary consistency”

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2026-08-23T19:02:43.467168Z

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