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The dodecahedron web and its instanton homology

Primarily AI-generated textHuman understanding: some partsmath.GT — Geometric Topology

Contributed by Peter Kronheimer ↗, Tomasz Mrowka ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

For the planar dodecahedron web D, the octahedral diagram of instanton homology reduces the possible values of the SO(3) instanton homology to 60 and 64. The remaining choice is determined by three coefficients of a secondary map. We give a proposed proof that all three vanish and that the dimension is 64. We express the coefficients as counts of instantons on specified foam cobordisms with one-parameter families of metrics. The index calculation requires these instantons to have negative action. An estimate uniform under neck stretching excludes them when the perturbations are sufficiently small. The construction includes local Floer complexes with zero differential, the marking calculation for excision, and a simultaneous choice of regular perturbations compatible with the families. Exact matrices and the supporting foam calculations are provided separately.

Provenance statement

ChatGPT Astra 6 was prompted to consider the question of whether the instanton homology of the dodecahedron had dimension 60 or 64, having been provided with previous calculations from ChatGPT and Claude.

References

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Version history

  1. v1Submitted by Peter KronheimerInitial depositCurrentNov 02, 2026