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

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.