Essential-Surface Complexes of Knot Exteriors: Image, Kernel, and Reconstruction

The curve complex of a surface is deliberately forgetful, yet in most cases its automorphism group recovers the mapping class group. We develop an analogous image–kernel–reconstruction viewpoint for the essential-surface complex $ES(E)$ of a knot exterior $E=E(K)$, equivalently Schultens’s initial surface complex $S_0(E)$. The paper is written as an entry point, beginning with explicit classical examples before introducing the general framework. We show that $ES(E)$ is a flag complex, that its boundary-bearing vertices are layered by boundary slope, and that although each $ES(E)$ is finite-dimensional, its dimension is unbounded over all knots. Kakimizu complexes of incompressible and minimal-genus spanning surfaces occur naturally as full subcomplexes. For slope-separated knots we determine the natural mapping-class-group action: every orientation-preserving mapping class acts trivially, while a nontrivial image occurs precisely through amphichirality. We illustrate the theory with torus knots, the figure-eight knot, cable knots, and connected sums; in the latter case annular spinning is already visible on $ES(E)$. We conclude with a recognition–realization–kernel roadmap, graded problems, and a worked two-bridge-knot recipe.