Knots, minimal surfaces and J-holomorphic curves

Fine, Joel

doi:10.2140/gt.2026.30.3207

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Knots, minimal surfaces and $J$-holomorphic curves

Joel Fine

Geometry & Topology 30 (2026) 3207–3292

DOI: 10.2140/gt.2026.30.3207

Abstract

Let $K$ be a knot in the 3-sphere, viewed as the ideal boundary of hyperbolic 4-space $ℍ^{4}$ . We prove that the number of minimal discs in $ℍ^{4}$ with ideal boundary $K$ is a knot invariant, i.e., the number is finite and doesn’t change under isotopies of $K$ . In fact, this gives a family of knot invariants, indexed by an integer describing the extrinsic topology of how the disc sits in $ℍ^{4}$ . These invariants can be seen as Gromov–Witten invariants counting $J$ -holomorphic discs in the twistor space $Z$ of $ℍ^{4}$ . Whilst Gromov–Witten theory suggests the general scheme for defining the invariants, there are substantial differences in how this must be carried out in our situation. These are due to the fact that the geometry of both $ℍ^{4}$ and $Z$ becomes singular at infinity, and so the $J$ -holomorphic curve equation is degenerate, rather than elliptic, at the boundary. This means that both the Fredholm and compactness arguments involve completely new features.

Keywords

knots, links, minimal surfaces, $J$-holomorphic curves, hyperbolic geometry, $0$-calculus

Mathematical Subject Classification

Primary: 53C42, 53C43, 53D45

Secondary: 57K10

References

Publication

Received: 30 November 2022

Revised: 2 February 2026

Accepted: 6 May 2026

Published: 12 October 2026

Proposed: Robert Lipshitz

Seconded: Leonid Polterovich, Ciprian Manolescu

Authors

Joel Fine
Départment de Mathématique Université libre de Bruxelles Brussels Belgium

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Volume 30, issue 9 (2026)