cs.LGFeb 19, 2026

Shortcut learning in geometric knot classification

Authors: Djordje Mihajlovic, Davide Michieletto

Organizations: School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, UK · School of Physics and Astronomy, University of Edinburgh, Edinburgh, EH9 3FD, UK · MRC Human Genetics Unit, Institute of Genetics and Cancer, University of Edinburgh, Edinburgh, EH4 2XU, UK · International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM2), Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan

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Abstract

Classifying the topology of closed curves is a central problem in low dimensional topology with applications beyond mathematics spanning protein folding, polymer physics and even magnetohydrodynamics. The central problem is how to determine whether two embeddings of a closed arc are equivalent under ambient isotopy. Given the striking ability of neural networks to solve complex classification tasks, it is therefore natural to ask if the knot classification problem can be tackled using Machine Learning (ML). In this paper, we investigate generic shortcut methods employed by ML to solve the knot classification challenge and specifically discover hidden non-topological features in training data generated through Molecular Dynamics simulations of polygonal knots that are used by ML to arrive to positive classifications results. We then provide a rigorous foundation for future attempts to tackle the knot classification challenge using ML by developing a publicly-available (i) dataset, that aims to remove the potential of non-topological feature classification and (ii) code, that can generate knot embeddings that faithfully explore chosen geometric state space with fixed knot topology. We expect that our work will accelerate the development of ML models that can solve complex geometric knot classification challenges.

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Figure 1: Shortcut learning of knot topology. A.) The unknot ( 010_{1} ) and the Conway knot ( 113411_{34} ) have the same (trivial) Alexander polynomial. However one cannot be deformed into the other. B.) A sketch of a neural network taking a knot embedding coordinates as input and outputting a binary unknot versus trefoil ( 010_{1} vs. 313_{1} ) classification. C.) An example of a configurational landscape of a knot embedding. The two main directions here are “size” and “writhe”, i.e. the amount of self-crossing of the curve. Molecular Dynamics simulations intrinsically bias sampling towards low free energy embeddings and therefore generate knot conformations with narrow distributions of geometric properties. D.) Example of a training dataset in which the ML takes an obvious shortcut learning based on the size of embeddings. All the trivial knots ( 010_{1} ) are smaller than the trefoils ( 313_{1} ) in the dataset. When the ML is challenged with 010_{1} that are large, and 313_{1} that are small it leads to incorrect classification. Figure layout inspired from [ 20 ] .
ϕj(x)\phi_{j}(\bm{x})Defn. (wrt. x\bm{x} )
Σ+\mathbf{\Sigma}_{+}∑i,j∣xi−xj∣\sum_{i,j}|\bm{x}_{i}-\bm{x}_{j}|
Ω+\mathbf{\Omega}_{+}∑i,j(dxi×dxj)⋅(xi−xj)/∣xi−xj∣3\sum_{i,j}(d\bm{x}_{i}\times d\bm{x}_{j})\cdot(\bm{x}_{i}-\bm{x}_{j})/|\bm{x}_{i}-\bm{x}_{j}|^{3}
κ+\bm{\kappa}_{+}∑iarccos((xi⋅xi+1)/(∣xi∣∣xi+1∣))\sum_{i}\text{arccos}((\bm{x}_{i}\cdot\bm{x}_{i+1})/(|\bm{x}_{i}||\bm{x}_{i+1}|))
MMax(∣xi−xj∣)\text{Max}(|\bm{x}_{i}-\bm{x}_{j}|)
Πn\mathbf{\Pi}_{n}∑i,jπij,πij∈Mij∣ connected component>n\sum_{i,j}\pi_{ij},\pi_{ij}\in M_{ij}|\text{ connected component}>n
Table 1: Functionals ( ϕj\phi_{j} ) used to identify potential shortcut learning. Here, xi=r(xi)\bm{x}_{i}=r(x_{i}) represents the 3D coordinates of vertex ii in the polygonal knot. From top to bottom: Σ+\mathbf{\Sigma}_{+} is the sum over pairwise distances of segments, Ω+\mathbf{\Omega}_{+} is the total writhe, κ+\bm{\kappa}_{+} is the total curvature, M is the largest pairwise distance and Πn\mathbf{\Pi}_{n} is the number of peaks in the pairwise distance matrix, such that the peaks are higher than the tolerance nn .
Figure 2: ML models fail to classify GEOKNOT dataset. A Examples of knot embeddings from MD simulations at low and high temperatures and of GEOKNOT. All three embeddings have 100 nodes. Plotted using KnotPlot [ 37 ] B Confusion matrices for models trained on MDLow T\text{MD}^{\text{Low T}} , MDHigh T\text{MD}^{\text{High T}} , and GEOKNOT respectively with coordinate data as the feature input. C Confusion matrices for models trained on MDLow T\text{MD}^{\text{Low T}} , MDHigh T\text{MD}^{\text{High T}} , and GEOKNOT respectively with writhe matrix data as the feature input.
Figure 3: MD datasets display limited sampling of geometries. Distributions of A total writhe, B average crossing number and C long-range entanglement for unknots ( 010_{1} ) and trefoil ( 313_{1} ) knots. From left to right we show distributions from MD low temp, high temp and GEOKNOT datasets. The definitions of these quantities are in Appendix B .
DATAΣ+\mathbf{\Sigma}_{+}Ω+\mathbf{\Omega}_{+}κ+\bm{\kappa}_{+}MΠ5\mathbf{\Pi}_{5}Π10\mathbf{\Pi}_{10}Π20\mathbf{\Pi}_{20}
MDLow T\text{MD}^{\text{Low T}}0.650.690.020.570.0830.530.45
MDHigh T\text{MD}^{\text{High T}}0.450.470.000.280.250.020.05
GEOKNOT0.020.030.050.010.020.010.00
Table 2: Mutual information reveals correlation between geometric features and knot type. Values of mutual information I(X;Y)I(X;Y) for different geometric features compared between MD low temp, high temp and GEOKNOT datasets. From left to right: sum of pairwise distances ( Σ+\Sigma_{+} ), sum of total space writhe ( Ω+\Omega_{+} ), total curvature ( κ+\kappa_{+} ), maximal pairwise distance ( M ), and the number of peaks on the pairwise matrix at varying tolerance ( Πn\Pi_{n} , n=5,10,20n=5,10,20 ). The high mutual information values for the MD datasets reflect the geometrically constrained conformational space that can be explored by the embeddings and suggest potential for shortcut learning.
DATACoordinate Acc.Writhe Acc.Shortcut Probes Acc.τcoord\bm{\tau}_{\textbf{coord}}τwr\bm{\tau}_{\textbf{wr}}
MD Low T\text{MD}^{\text{ Low T}}99.999.999.91.001.00
MD High T\text{MD}^{\text{ High T}}99.996.783.10.830.86
GEOKNOT49.967.262.3-0.93
Table 3: ML classification accuracy reveals shortcut learning. Summary of models and their best accuracies for various input data. The dramatic drop in accuracy for GEOKNOT reflects the models inability to classify topology of curves with broad distributions in geometric features. The “shortcut probes acc.” column reports accuracy of ML models solely trained on the shortcut probes identified by our algorithm. Notice that τcoord\bm{\tau}_{coord} is omitted for GEOKNOT data because the coordinate model accuracy is random ( ≃50%\simeq 50\% ), meaning τ\tau is not informative about shortcut reliance in this regime.
Figure 4: Saliency analysis identify key geometric probes for correct classification . Saliency analysis on ML models trained on geometric probes using both low (A) and high (B) temperature MD (LAMMPS) datasets. At low temperature the Ω+\Omega_{+} feature is by far the most important for correct classification, suggesting that total space writhe is a dominant shortcut feature.
Figure 5: ML model prediction is not invariant under ambient isotopy. In A and B we show two examples of 010_{1} embeddings generated by GEOKNOT that are misclassified as 313_{1} by ML models trained on MD data. The embeddings are minimised by KnotPlot until all non-topological self-crossings are removed. In A the model is trained on the XYZ coordinates of MD data. In B the model is trained on the writhe matrix of MD data. Note, in both plots a smoothed plot of the knot embedding is provided for visual clarity.
MODELSleiman et. al. [ 17 ]Braghetto et. al. [ 19 ]Zhang et. al. [ 18 ]
REPRESENTATIONWrithe matrixCoordinatesCoordinates & bond vectors
LAMMPS Low T\text{LAMMPS}^{\text{ Low T}}100.0100.097.7
LAMMPS High T\text{LAMMPS}^{\text{ High T}}96.7100.097.2
GEOKNOT52.331.614.4
Table 4: Testing existing ML models for knot classification. Classification accuracy of ML models in current literature against our MD (LAMMPS) datasets and GEOKNOT dataset.
Figure 6: v2v_{2} approximation from writhe matrix over the low and high temperature LAMMPS and GEOKNOT datasets using writhe matrix data.
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