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 ℰ𝒮(E) of a knot exterior E=E(K), equivalently Schultens's initial surface complex 𝒮0(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 S2× S1, 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 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.
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BibTeX
@misc{ozawa2026essentialsurface,
title = {{Essential-Surface Complexes of Knot Exteriors: Image, Kernel, and Reconstruction}},
author = {Makoto Ozawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.20869},
eprint = {2607.20869},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.20869},
}
RIS
TY - UNPB
TI - Essential-Surface Complexes of Knot Exteriors: Image, Kernel, and Reconstruction
AU - Makoto Ozawa
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2607.20869
UR - https://arxiv.org/abs/2607.20869
AB - 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 ℰ𝒮(E) of a knot exterior E=E(K), equivalently Schultens's initial surface complex 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× 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 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.
ER -