A new mathematical preprint finds that the full link Floer complex of the three-component Borromean rings is not homotopy equivalent to a free resolution of its homology. The result, known as nonformality, marks a precise difference between the complex and the simpler algebraic object used to represent its homology.
The authors also construct an explicit complex that is homotopy equivalent to the Borromean-rings Floer complex, giving researchers a concrete model for working with it. In this setting, homotopy equivalence means that the two complexes represent the same object for the purposes of the theory, even though their algebraic descriptions may differ.
A difference that matters
The central evidence is a rank mismatch. The free resolution calculated for the Borromean-rings Floer homology has rank 66, while the Floer homology itself has rank 64. Because the two ranks do not agree, the full complex cannot be homotopy equivalent to that free resolution of its homology.
The paper studies the link Floer complex as a free, finitely generated chain complex over a polynomial ring with six variables. The candidate model is built by removing two generators from a tensor product, then adding two new generators with a correction to the differential, the rule that connects one part of the complex to another.
That construction is not presented as a numerical estimate or an experimental observation. It is an algebraic computation and proof about the Borromean rings, with the free-resolution calculation supported by Macaulay2 code included in the paper.
A criterion, and a controlled exception
The authors use the example to state a broader test for L-space links. For an L-space link in S3, vanishing of the specified graded Ext groups in every degree above 1 is sufficient to establish that the link Floer complex is homotopy equivalent to a free resolution of its homology. Ext groups are algebraic measurements used here to test whether additional structure remains beyond that resolution.
For the Borromean rings, the reported Ext calculation does not meet that vanishing condition: the relevant graded Ext group is one-dimensional in degree 3 and zero in the other reported degrees above 1. The paper also finds exactly one nontrivial deformation of the free-resolution complex up to equivalence.
In plain terms, the result points to one specific algebraic way in which the Borromean-rings complex differs from its free resolution. The authors interpret that nontrivial deformation as the feature that accounts for the complex’s nonformality.
Eight models, with one assignment still open
The preprint also examines the link involution, another algebraic structure in the problem. Up to equivalence, it finds eight distinct algebraic models, associated with choices of twisting directions.
That count comes with an important qualification. The authors do not determine which twisting direction produces each individual involution model, leaving the correspondence between the choices and the algebraic models unresolved.
The result’s boundaries
The paper identifies the Borromean rings as an L-space link and states its formality criterion in that setting. The specific calculations, however, concern the Borromean rings; they do not by themselves establish the same behavior for unrelated links.
The findings are theoretical rather than empirical. There is no human, animal, or laboratory sample, and the paper reports no statistical analysis. The supplied document also notes that the computer-assisted free-resolution and Ext calculations are based on Macaulay2 output, without independent verification or complete computational logs beyond the code excerpts.
The paper is an arXiv version 1 preprint dated 26 August 2026, with no journal information reported in the supplied record. It reports partial support from NSF grants, a Simons Travel Fellowship, and a Sloan Fellowship.
Questions left by the calculation
The unresolved involution assignment is one open question. The authors also ask whether working over the integers could remove the remaining ambiguity, and how broadly deformation and Ext-based tests can classify non-plumbed L-space links with more than two components.
Paper data and sources
Original title: Link Floer homology, nonformality, and the Borromean rings
Authors: Kristen Hendricks, Matthew Stoffregen, Ian Zemke
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text