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

Abstract

This is an expository article. The curve complex of a surface is deliberately forgetful, yet in most cases its automorphism group recovers the mapping class group. We explain what happens when the same question is asked for the essential-surface complex $\mathcal{E}\mathcal{S}(E)$ of a knot exterior $E=E(K)$, equivalently Schultens's initial surface complex ${\mathcal{S}}_0(E)$. The discussion is organized around three questions: which combinatorial symmetries are geometric (the image), which homeomorphisms are invisible (the kernel), and how much of $E$ can be read off from the complex (reconstruction). The basic objects are first drawn in classical examples, and the general framework is introduced only afterwards. We collect the elementary structural facts that organize the subject -- the flag property, the layering by boundary slope, finite dimension, unbounded dimension for iterated cables, and Kakimizu's complexes as full subcomplexes -- and compare with handlebodies and connected sums of $S^2\times S^1$, where image and kernel are known completely. Along the way we record two short consequences that we have not found stated in this form: for a slope-separated knot every orientation-preserving mapping class acts trivially, and the image is $\mathbb{Z}/2$ exactly when the knot is amphichiral and a nonzero boundary slope occurs; and for a connected sum of nonfibered prime knots the decomposing-annulus twist translates Banks's winding coordinate. A table separates what is classical from what is recorded here. The article closes with a three-task formulation of the rigidity problem, a list of problems with their current status, a worked two-bridge recipe, and a guide to further reading.

arXiv comment41 pages, 17 figures, 1 table. v3: rewritten as an expository article; added a status table of classical vs. new results, a comparison with handlebodies and #S^2xS^1, a problem list with status, and further references; corrected the two-bridge continued-fraction convention