Holomorphic Disk Counts Yield New Obstructions via Lagrangian Knot Cobordisms — E8 Intelligence Research

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-14
DOI
https://doi.org/10.5281/zenodo.22742286
Primary Topic
Geometric and Algebraic Topology
Type
preprint
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preprint

Holomorphic Disk Counts Yield New Obstructions via Lagrangian Knot Cobordisms — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Geometric and Algebraic Topology
preprint

Holomorphic Disk Counts Yield New Obstructions via Lagrangian Knot Cobordisms — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: Lagrangian cobordisms between enriched knot diagrams yield new algebraic obstructions via holomorphic disk counts, linking knot theory to symplectic topology through moduli spaces of pseudoholomorphic curves. | MATH: The core invariant arises from the Euler characteristic of the moduli space of holomorphic disks with boundary on Lagrangian submanifolds in \(\mathbb{R}^4\). For a Lagrangian cobordism \(L \subset \mathbb{R}^4 \times \mathbb{R}\) between knots \(K_-\) and \(K_+\), the count of holomorphic disks (with Maslov index 2) gives a map \(\Phi: \mathcal{L} \to \mathbb{Z}\) satisfying \(\Phi(K_+) - \Phi(K_-) = \chi(\mathcal{M})\), where \(\mathcal{M}\) is the disk moduli. Enriched knot diagrams encode crossings with signs, and the obstruction depends on the "writhe" and "self-linking" numbers — specifically, the difference in the \(sl\)-invariant (self-linking) between boundary knots must vanish for a Lagrangian cobordism, a condition expressible as \(\Delta sl = 0\). The Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
Geometric and Algebraic Topology
DOIOpen Access PDFOpenAlex
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Holomorphic Disk Counts Yield New Obstructions via Lagrangian Knot Cobordisms — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS