Learning Topological Features of $\widehat Z$-invariants

Brandon Robinson; Shimal Harichurn; Fabian Ruehle; Sergei Gukov; Rak-Kyeong Seong; Miranda C. N. Cheng

hep-thAug 19, 2026

Learning Topological Features of $\widehat Z$ -invariants

Authors: Brandon Robinson, Shimal Harichurn, Fabian Ruehle, Sergei Gukov, Rak-Kyeong Seong, Miranda C. N. Cheng

Organizations: Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands · Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 105-107, 1098 XG Amsterdam, Netherlands · School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, South Africa · Department of Physics, Northeastern University, Boston, MA 02115, USA · Department of Mathematics, Northeastern University, Boston, MA 02115, USA · The NSF AI Institute for Artificial Intelligence and Fundamental Interactions · Richard N. Merkin Center for Pure and Applied Mathematics, California Institute of Technology, Pasadena, CA 91125, USA · Department of Mathematical Sciences, and Department of Physics, Ulsan National Institute of Science and Technology, 50 UNIST-gil, Ulsan 44919, South Korea · Institute for Mathematics, Academia Sinica, Taipei 106319, Taiwan

Abstract

Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology. In this paper, we initiate a systematic approach to handling mathematical data structured as (truncated) infinite $q$ -series, or equivalently, infinite series of integers. To apply this data analysis pipeline, we construct a comprehensive dataset of $\widehat{Z}$ -invariants (homological blocks) for plumbed 3-manifolds. We demonstrate that neural networks can reliably extract essential topological information, such as homology class and underlying graph structure, directly from the $q$ -series coefficients. A central feature of our methodology is a focus on interpretability; by contrasting local gradient sensitivity with global feature relevance, we reveal that the networks learn to bypass complex topological rules in favor of specific spectral and geometric proxies. Finally, we apply this pipeline to probe homology cobordism, discovering a high-accuracy predictive relationship between the $\widehat{Z}$ -invariant exponents and the Heegaard Floer $d$ -invariant (correction term). These results suggest that $\widehat{Z}$ -invariants capture subtle geometric information regarding cobordism equivalences, warranting a new direction for the study of quantum invariants.

Figures & tables

Panel of Figure 1 — Figure 1: Schematic of the ML-assisted data analysis pipeline.

	Homology Sphere	Non-Sphere
Star	13228	31706
H-shaped	7952	10717

Table 1: Summary of datasets split between homology and graph topology labels used for binary classification tasks.

$\#$ singular fibers	3	4	5	6	7
$\#$ samples	6000	5998	5988	5984	5978

Table 2: Data for multi-class experiments learning the number of singular fibers of Seifert manifolds from

\widehat{Z}

Panel of Figure 2 — Figure 2: A star-shaped graph representing a Seifert manifold. Vertex labels have been omitted.

Panel of Figure 3 — Figure 3: A generic H-shaped plumbing graph.

Panel of Figure 4 — Figure 4: A splice diagram with two nodes and four leaves.

Stage	Input	Architecture	Batch	Learning rate
Classifier	Full Sequence	$(2,64)$	$128$	$10^{-3}$
Classifier	Value-Index	$(2,64)$	$128$	$10^{-3}$
Contrastive encoder	Full Sequence	$(4,128)$	$128$	$10^{-3}$
CL classifier	Latent Space	$(2,32)$	$128$	$10^{-3}$
Regressor	Abstract features	$(4,128)$	$64$	$10^{-4}$

Table 3: Summary of the network architectures and hyperparameters used in the seed-robust topology and homology experiments. The architecture column records

(d,w)

, where

d

is the number of affine layers and

w

is the hidden-layer width. Classifiers are trained with cross-entropy loss and selected by balanced accuracy. The contrastive encoder is trained with semi-hard triplet loss, and the regressor is trained with mean-squared error to reconstruct the scaled classifier logits from the ten abstract features.

Experiment	BAS	$\mathbf{R^{2}}$	PCC	FS	PI	SHAP
HOM-Star	$0.928\,\pm\,0.003$	$\mathrm{tr}:0.430\,\pm\,0.253$ $\mathrm{val}:0.193\,\pm\,0.290$	$\mathrm{tr}:0.885\,\pm\,0.123$ $\mathrm{val}:0.565\,\pm\,0.476$	$\varepsilon_{\rm min}$ $1.8$ , $\sigma_{\gamma}$ $4.1$ , $\Delta$ $4.1$	$\varepsilon_{\rm min}$ $2.3$ , $\sigma_{\varepsilon}$ $3.2$ , $\varepsilon_{\rm max}$ $4.3$	$\varepsilon_{\rm min}$ $1.4$ , $\sigma_{\varepsilon}$ $3.1$ , $\varepsilon_{\rm max}$ $4.4$
HOM-H	$0.923\,\pm\,0.009$	$\mathrm{tr}:0.769\,\pm\,0.023$ $\mathrm{val}:0.396\,\pm\,0.039$	$\mathrm{tr}:0.879\,\pm\,0.014$ $\mathrm{val}:0.742\,\pm\,0.013$	$\mu_{\gamma}$ $1.3$ , $\sigma_{\gamma}$ $1.7$ , $\varepsilon_{\rm min}$ $3.5$	$\sigma_{\gamma}$ $1.7$ , $\mu_{\gamma}$ $3.4$ , $\varepsilon_{\rm min}$ $3.8$	$\sigma_{\gamma}$ $2.0$ , $\varepsilon_{\rm min}$ $2.2$ , $\mu_{\gamma}$ $2.5$
HOM-All	$0.952\,\pm\,0.003$	$\mathrm{tr}:0.698\,\pm\,0.035$ $\mathrm{val}:0.572\,\pm\,0.029$	$\mathrm{tr}:0.852\,\pm\,0.025$ $\mathrm{val}:0.798\,\pm\,0.026$	$\det$ $1.5$ , $\varepsilon_{\rm min}$ $2.1$ , $c_{\rm max}$ $3.6$	$\sigma_{\varepsilon}$ $2.5$ , $\varepsilon_{\rm max}$ $2.8$ , $\varepsilon_{\rm min}$ $3.8$	$\varepsilon_{\rm min}$ $1.4$ , $\det$ $2.0$ , $\sigma_{\gamma}$ $3.1$
TOP-Sph	$0.964\,\pm\,0.004$	$\mathrm{tr}:0.837\,\pm\,0.012$ $\mathrm{val}:0.725\,\pm\,0.017$	$\mathrm{tr}:0.916\,\pm\,0.007$ $\mathrm{val}:0.863\,\pm\,0.010$	$\mu_{\gamma}$ $1.0$ , $c_{\rm max}$ $2.5$ , $\sigma_{\gamma}$ $3.6$	$\mu_{\gamma}$ $1.0$ , $\sigma_{\gamma}$ $2.0$ , $\Delta$ $3.4$	$\mu_{\gamma}$ $1.0$ , $\sigma_{\gamma}$ $2.0$ , $\det$ $3.6$
TOP-Non	$0.971\,\pm\,0.009$	$\mathrm{tr}:0.759\,\pm\,0.019$ $\mathrm{val}:0.600\,\pm\,0.029$	$\mathrm{tr}:0.876\,\pm\,0.010$ $\mathrm{val}:0.819\,\pm\,0.012$	$\det$ $1.0$ , $\varepsilon_{\rm min}$ $2.9$ , $\sigma_{\gamma}$ $3.5$	$\varepsilon_{\rm min}$ $2.1$ , $\det$ $2.8$ , $\mu_{\varepsilon}$ $3.7$	$\det$ $1.0$ , $\varepsilon_{\rm min}$ $2.4$ , $\sigma_{\gamma}$ $3.8$
TOP-All	$0.978\,\pm\,0.004$	$\mathrm{tr}:0.761\,\pm\,0.018$ $\mathrm{val}:0.616\,\pm\,0.013$	$\mathrm{tr}:0.876\,\pm\,0.010$ $\mathrm{val}:0.816\,\pm\,0.010$	$\det$ $1.0$ , $\varepsilon_{\rm min}$ $2.2$ , $\sigma_{\gamma}$ $3.0$	$\varepsilon_{\rm min}$ $2.6$ , $\sigma_{\gamma}$ $3.0$ , $\sigma_{\varepsilon}$ $3.3$	$\sigma_{\gamma}$ $1.6$ , $\varepsilon_{\rm min}$ $2.1$ , $\det$ $2.3$

Table 4: Multi-seed full-sequence results for the topology and homology experiments. BAS is reported as mean

\pm

standard deviation over classifier seeds. The

R^{2}

and Pearson correlation (PCC) columns report train and validation reconstruction quality of the feature-to-logit regressor, stacked within each cell and averaged over the classifier–regressor seed ensemble. The final columns report the three lowest mean feature ranks for feature saliency (FS), permutation importance (PI), and SHAP, where the subscript gives the average rank over the normalized interpretability ensemble.

Panel of Figure 5 — Figure 5: Two plumbings of $\Sigma(2,3,5)$ equivalent under Kirby–Neumann moves.

Feature	Skewness	Kurtosis
$\det$	0.00	17.73
$\varepsilon_{\rm min}$	1.65	9.98
$\varepsilon_{\rm max}$	5.15	45.54
$\mu_{\varepsilon}$	2.01	10.48
$\sigma_{\varepsilon}$	5.40	50.47
$\Delta$	10.93	148.67

Table 5: Higher moments of the ten-feature distribution. Skewness and kurtosis are computed on the raw feature matrix used in the topology experiments, before the interpretability scores are ranked. Large values indicate heavy-tailed or strongly non-normal feature marginals.

Experiment	FS	PI	SHAP
HOM-Star	$-0.200\,\pm\,0.416$	$-0.315\,\pm\,0.184$	$-0.425\,\pm\,0.190$
HOM-H	$-0.481\,\pm\,0.100$	$-0.456\,\pm\,0.137$	$-0.577\,\pm\,0.117$
HOM-All	$-0.194\,\pm\,0.157$	$-0.505\,\pm\,0.142$	$-0.709\,\pm\,0.115$
TOP-Sph	$0.304\,\pm\,0.200$	$0.201\,\pm\,0.122$	$-0.017\,\pm\,0.176$
TOP-Non	$-0.391\,\pm\,0.253$	$-0.769\,\pm\,0.111$	$-0.761\,\pm\,0.058$
TOP-All	$-0.362\,\pm\,0.064$	$-0.666\,\pm\,0.174$	$-0.769\,\pm\,0.028$

Table 6: Multi-seed higher-moment bias scores for the full-sequence interpretability metrics. Entries are reported as mean

\pm

standard deviation over the classifier–regressor–interpretability seed ensemble. Positive values indicate that an attribution method preferentially ranks features with large skewness or kurtosis, while negative values indicate preference for features with more stable marginals. Scores are computed via ( 10 ) from Spearman correlations between each metric’s feature-importance vector and the feature-moment rankings of Tab. 5 .

Task	Student feats.	$B_{10}$ feats.	Full	$B_{5}$	$B_{10}$	$B_{20}$	$E_{50}$
HOM-Star	$\varepsilon$ , log.	$\varepsilon_{\min},\,\sigma_{\varepsilon}$	$.726\pm.258$	$0.584\pm 0.075$	$.604\pm.127$	$.623\pm.177$	$.762\pm.237$
HOM-H	$\det$ , RF	$\det$	$.920\pm.007$	$0.576\pm 0.037$	$.633\pm.015$	$.716\pm.032$	$.973\pm.011$
HOM-All	$\det$ , RF	$\det$	$.953\pm.005$	$0.737\pm 0.021$	$.809\pm.014$	$.869\pm.017$	$.970\pm.005$
TOP-Sph	$\gamma$ , log.	$\mu_{\gamma},\,\gamma_{0}$	$.969\pm.005$	$0.682\pm 0.040$	$.762\pm.030$	$.850\pm.026$	$.990\pm.003$
TOP-Non	$\det+\gamma$ , MLP	$\sigma_{\gamma},\,\mu_{\gamma}$	$.986\pm.003$	$0.861\pm 0.035$	$.909\pm.029$	$.932\pm.028$	$.985\pm.005$
TOP-All	$\gamma$ , log.	$\sigma_{\gamma},\,\mu_{\gamma}$	$.977\pm.002$	$0.784\pm 0.013$	$.856\pm.011$	$.905\pm.011$	$.985\pm.003$

Table 7: Normalized margin reconstruction for the full-sequence teachers, with feature attribution on the

B_{10}

teacher-boundary slice. The source teachers and train/validation splits are inherited directly from the full-sequence experiments. Students are trained to reproduce the source teacher’s hard decisions using only the indicated abstract feature set. “Full” reports balanced agreement with the source teacher on the held-out split.

B_{5}

B_{10}

and

B_{20}

restrict agreement to the 5%, 10%, and 20% smallest-margin held-out samples, where the margin is

|\ell_{1}-\ell_{0}|

E_{50}

reports agreement on the 50% largest-margin held-out samples. “

B_{10}

feats.” lists the leading features under the average of normalized saliency, SHAP, and permutation-drop attribution scores. Here

\varepsilon

-stats denotes

(\varepsilon_{\min},\varepsilon_{\max},\mu_{\varepsilon},\sigma_{\varepsilon})

, and

\gamma

-stats denotes

(\gamma_{0},\mu_{\gamma},\sigma_{\gamma})

Experiment	BAS	$\mathbf{R^{2}}$	PCC	FS	PI	SHAP
HOM-Star	$0.999\,\pm\,0.001$	$\mathrm{tr}:0.940\,\pm\,0.013$ $\mathrm{val}:0.920\,\pm\,0.018$	$\mathrm{tr}:0.969\,\pm\,0.007$ $\mathrm{val}:0.958\,\pm\,0.011$	$\Delta$ $1.3$ , $\mu_{\gamma}$ $1.7$ , $\det$ $3.7$	$\mu_{\gamma}$ $1.2$ , $\varepsilon_{\rm min}$ $2.0$ , $\det$ $3.5$	$\varepsilon_{\rm min}$ $1.4$ , $\mu_{\gamma}$ $1.6$ , $\det$ $3.0$
HOM-H	$0.967\,\pm\,0.004$	$\mathrm{tr}:0.971\,\pm\,0.023$ $\mathrm{val}:0.951\,\pm\,0.031$	$\mathrm{tr}:0.986\,\pm\,0.011$ $\mathrm{val}:0.976\,\pm\,0.015$	$\mu_{\gamma}$ $1.0$ , $\sigma_{\gamma}$ $2.0$ , $\Delta$ $3.0$	$\mu_{\gamma}$ $1.7$ , $\Delta$ $2.5$ , $\sigma_{\gamma}$ $2.8$	$\mu_{\gamma}$ $1.0$ , $\sigma_{\gamma}$ $2.0$ , $\det$ $3.3$
HOM-All	$0.985\,\pm\,0.002$	$\mathrm{tr}:0.892\,\pm\,0.029$ $\mathrm{val}:0.851\,\pm\,0.030$	$\mathrm{tr}:0.948\,\pm\,0.014$ $\mathrm{val}:0.927\,\pm\,0.013$	$\det$ $1.0$ , $\varepsilon_{\rm min}$ $2.0$ , $\sigma_{\gamma}$ $3.5$	$\sigma_{\varepsilon}$ $2.2$ , $\varepsilon_{\rm min}$ $2.4$ , $\mu_{\gamma}$ $3.3$	$\varepsilon_{\rm min}$ $1.5$ , $\det$ $2.3$ , $\mu_{\gamma}$ $2.6$
TOP-Sph	$1.000\,\pm\,0.000$	$\mathrm{tr}:0.990\,\pm\,0.002$ $\mathrm{val}:0.987\,\pm\,0.004$	$\mathrm{tr}:0.995\,\pm\,0.001$ $\mathrm{val}:0.994\,\pm\,0.002$	$\mu_{\gamma}$ $1.0$ , $\sigma_{\gamma}$ $2.2$ , $\gamma_{0}$ $3.0$	$\mu_{\gamma}$ $1.1$ , $\sigma_{\gamma}$ $2.0$ , $\gamma_{0}$ $3.1$	$\mu_{\gamma}$ $1.0$ , $\sigma_{\gamma}$ $2.1$ , $\Delta$ $3.0$
TOP-Non	$1.000\,\pm\,0.000$	$\mathrm{tr}:0.979\,\pm\,0.003$ $\mathrm{val}:0.975\,\pm\,0.004$	$\mathrm{tr}:0.990\,\pm\,0.002$ $\mathrm{val}:0.988\,\pm\,0.002$	$\mu_{\gamma}$ $1.4$ , $\Delta$ $3.2$ , $\det$ $3.6$	$\mu_{\gamma}$ $1.4$ , $\sigma_{\gamma}$ $2.3$ , $\varepsilon_{\rm min}$ $3.7$	$\mu_{\gamma}$ $1.0$ , $\varepsilon_{\rm min}$ $2.2$ , $\sigma_{\gamma}$ $3.3$
TOP-All	$1.000\,\pm\,0.000$	$\mathrm{tr}:0.960\,\pm\,0.003$ $\mathrm{val}:0.952\,\pm\,0.005$	$\mathrm{tr}:0.980\,\pm\,0.002$ $\mathrm{val}:0.977\,\pm\,0.002$	$\Delta$ $1.8$ , $\sigma_{\gamma}$ $2.5$ , $\varepsilon_{\rm min}$ $3.8$	$\varepsilon_{\rm min}$ $2.3$ , $\mu_{\gamma}$ $2.6$ , $\sigma_{\gamma}$ $3.1$	$\sigma_{\gamma}$ $1.0$ , $\mu_{\gamma}$ $2.1$ , $\varepsilon_{\rm min}$ $3.0$

Table 8: Multi-seed value-index results for the topology and homology experiments. BAS is reported as mean

\pm

standard deviation over classifier seeds. The

R^{2}

Feature	TOP coefficient	HOM coefficient
$\det$	$0.03\,\pm\,0.01$	$2.17\,\pm\,0.04$
$\varepsilon_{\rm min}$	$0.30\,\pm\,0.01$	$4.43\,\pm\,0.16$
$\varepsilon_{\rm max}$	$-0.27\,\pm\,0.04$	$-0.62\,\pm\,0.03$
$\mu_{\varepsilon}$	$0.16\,\pm\,0.02$	$-0.26\,\pm\,0.02$
$\sigma_{\varepsilon}$	$0.19\,\pm\,0.06$	$0.63\,\pm\,0.04$
$\Delta$	$1.13\,\pm\,0.03$	$-0.35\,\pm\,0.04$

Table 9: Multi-seed normalized coefficients

\kappa_{i}

for the linear abstract-feature boundary approximation

\sum_{i}\kappa_{i}f_{i}=0

. The TOP column is obtained from a class-balanced linear classifier separating graph topology classes, while the HOM column is obtained from a class-balanced linear classifier separating homology spheres from non-spheres. For each seed, the ten abstract features are standardized on the training split, the linear classifier is fit, and the coefficient vector is rescaled so that

\frac{1}{10}\sum_{i}|\kappa_{i}|=1

. Entries report mean

\pm

standard deviation over the five-seed ensemble. Coefficient signs follow the binary label encoding, with positive coefficients pointing toward the star-shaped class (TOP row) and the non-sphere class (HOM row); coefficient magnitudes should be interpreted as robust linear feature strengths.

Figure 6: One-vs-rest decision-boundary reconstruction in the four-class contrastive latent representation. Columns correspond to the four classes STAR-Non, STAR-Sph, H-Sph, and H-Non. The top row shows the seed - $42$ latent-space projection with an RBF-SVM one-vs-rest decision function: colored points are the positive class, gray points are the complementary classes, filled contours show the SVM decision value, and the dashed black curve is the zero-level decision boundary. The middle row reports the multi-seed mean feature importances of the cross-validation-selected reconstruction model for the corresponding boundary, with error bars showing seed-to-seed standard deviation. The bottom row reports multi-seed reconstruction quality using $R^{2}$ , Spearman correlation $\rho$ , sign accuracy $\vartheta$ , and boundary agreement $\mathcal{S}_{\rm MAE}$ . Together, the panels show that the latent one-vs-rest boundaries are largely reconstructible from the abstract features, with STAR-side boundaries emphasizing $\varepsilon_{\rm min}$ and $\det$ , and H-side boundaries emphasizing gap-statistic features such as $\mu_{\gamma}$ and $\sigma_{\gamma}$ . The STAR-Non boundary is the least stable across seeds and accounts for most of the variance in the reconstruction diagnostics.

OvR Boundary	Best Model	Best FI	$\mathbf{Q_{\mathrm{composite}}}$
STAR-Non	RandomForest	$\mathbf{\varepsilon_{\rm min}}$ $0.342\pm 0.049$ , $\mathbf{\det}$ $0.260\pm 0.027$ , $\mathbf{\mu_{\gamma}}$ $0.147\pm 0.037$	$0.898\,\pm\,0.201$
STAR-Sph	GradBoost	$\mathbf{\varepsilon_{\rm min}}$ $0.304\pm 0.241$ , $\mathbf{\mu_{\gamma}}$ $0.220\pm 0.168$ , $\mathbf{\det}$ $0.211\pm 0.142$	$0.975\,\pm\,0.013$
H-Sph	RandomForest	$\mathbf{\mu_{\gamma}}$ $0.462\pm 0.276$ , $\mathbf{\sigma_{\gamma}}$ $0.218\pm 0.125$ , $\mathbf{\gamma_{0}}$ $0.113\pm 0.065$	$0.981\,\pm\,0.009$
H-Non	GradBoost	$\mathbf{\mu_{\gamma}}$ $0.569\pm 0.337$ , $\mathbf{\varepsilon_{\rm min}}$ $0.182\pm 0.306$ , $\mathbf{\sigma_{\gamma}}$ $0.108\pm 0.131$	$0.963\,\pm\,0.023$

Table 10: Multi-seed reconstruction summary for the one-vs-rest decision boundaries in the four-class contrastive latent space. The top-feature column reports the three largest feature importances of the cross-validation-selected reconstruction model, with mean

\pm

standard deviation over seeds displayed below each feature. The composite score

Q_{\mathrm{composite}}

is the weighted combination of the four reconstruction diagnostics defined in Appendix B . All entries are averaged over the seed ensemble.

Experiment	BAS	FS	PI	SHAP
HOM-Star	$1.000\,\pm\,0.000$	$\varepsilon_{\rm min}$ $1.0$ , $\mu_{\gamma}$ $2.2$ , $\Delta$ $2.8$	$\varepsilon_{\rm min}$ $1.0$ , $\mu_{\gamma}$ $2.0$ , $\det$ $3.6$	$\varepsilon_{\rm min}$ $1.0$ , $\mu_{\gamma}$ $2.0$ , $\sigma_{\varepsilon}$ $3.3$
HOM-H	$0.998\,\pm\,0.001$	$\varepsilon_{\rm min}$ $1.2$ , $\det$ $1.8$ , $\sigma_{\gamma}$ $3.0$	$\varepsilon_{\rm min}$ $1.0$ , $\det$ $2.0$ , $\sigma_{\gamma}$ $3.4$	$\varepsilon_{\rm min}$ $1.4$ , $\det$ $1.6$ , $\sigma_{\gamma}$ $3.2$
HOM-All	$0.995\,\pm\,0.001$	$\varepsilon_{\rm min}$ $1.0$ , $\Delta$ $2.2$ , $\sigma_{\gamma}$ $2.8$	$\varepsilon_{\rm min}$ $1.0$ , $\varepsilon_{\rm max}$ $2.5$ , $\sigma_{\gamma}$ $3.6$	$\varepsilon_{\rm min}$ $1.0$ , $\sigma_{\gamma}$ $2.0$ , $\varepsilon_{\rm max}$ $4.1$
TOP-Sph	$1.000\,\pm\,0.000$	$\mu_{\gamma}$ $1.0$ , $\sigma_{\gamma}$ $2.0$ , $\Delta$ $3.4$	$\mu_{\gamma}$ $1.0$ , $\sigma_{\gamma}$ $2.0$ , $c_{\rm max}$ $3.4$	$\mu_{\gamma}$ $1.0$ , $\sigma_{\gamma}$ $2.0$ , $\varepsilon_{\rm max}$ $3.5$
TOP-Non	$1.000\,\pm\,0.000$	$\varepsilon_{\rm min}$ $1.0$ , $\gamma_{0}$ $2.0$ , $\Delta$ $3.4$	$\varepsilon_{\rm min}$ $1.0$ , $\varepsilon_{\rm max}$ $2.6$ , $\sigma_{\gamma}$ $2.6$	$\varepsilon_{\rm min}$ $1.0$ , $\sigma_{\varepsilon}$ $2.4$ , $\sigma_{\gamma}$ $2.8$
TOP-All	$1.000\,\pm\,0.000$	$\sigma_{\gamma}$ $1.0$ , $\gamma_{0}$ $2.0$ , $\mu_{\gamma}$ $3.2$	$\sigma_{\gamma}$ $1.0$ , $\mu_{\gamma}$ $2.0$ , $\varepsilon_{\rm min}$ $3.0$	$\sigma_{\gamma}$ $1.0$ , $\mu_{\gamma}$ $2.1$ , $\varepsilon_{\rm min}$ $3.5$

Table 11: Multi-seed direct feature-classifier results. The classifier input is the ten-feature vector

(\det,\varepsilon_{\min},\varepsilon_{\max},\mu_{\varepsilon},\sigma_{\varepsilon},\Delta,\gamma_{0},\mu_{\gamma},\sigma_{\gamma},c_{\max})

rather than the truncated

\widehat{Z}

sequence. BAS is reported as mean

\pm

standard deviation over classifier seeds. The final columns report the three lowest mean feature ranks for feature saliency (FS), permutation importance (PI), and SHAP, with the average rank displayed below each feature.

	$r=3$	$r=4$	$r=5$
Poly(2), offsets only (best over $L$ )	$1.0000\pm 0.0000$	$1.0000\pm 0.0000$	$0.9950\pm 0.0019$
Poly(2), offsets $+\,\Delta$	$1.0000\pm 0.0000$	$1.0000\pm 0.0000$	$1.0000\pm 0.0000$

Table 12: Cross-validated

R^{2}

for a degree-2 polynomial regressing

d+\Delta-\tfrac{1}{2}

on the

\widehat{Z}

exponent offsets of Seifert fibered integer homology spheres with

r

exceptional fibers (

n=3020,1743,306

for

r=3,4,5

). The best-over-

L

scores in the offsets-only row are attained at

L=6

12

, and

16

for

r=3,4,5

respectively. The

+\Delta

entries are evaluated at fixed base prefix lengths

L=4,7,10

for

r=3,4,5

: the empirical Poly(2)

R^{2}\geq 0.99

crossings for

r=3,4

, with

L=10

chosen for

r=5

, whose crossing is unstable at

n=306

. These evaluation defaults are distinct from the

99.9\%

thresholds

L_{99.9\%}

discussed in the text.

Figure 7: $SL_{2}(\mathbb{Z})$ -orbit regression: attribution diagnostics for the flat-block transformer. The input is ordered as $[c_{0},\ldots,c_{63},e_{0},\ldots,e_{63}]$ . The four panels show gradient saliency, integrated gradients, gradient-times-input, and attention rollout, averaged over the $30$ -seed robustness ensemble after normalizing each seed by its total attribution mass. The dashed red line separates the coefficient block from the exponent block. Bar color denotes the within-block index modulo $8$ , reflecting positional organization associated with the eight interleaved partial-theta sectors in the Brieskorn-sphere expression.

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Model	STAR-Sph	STAR-Non	H-Sph	H-Non	Average
Random Forest	$\mathbf{0.966\,\pm\,0.020}$	$\mathbf{0.977\,\pm\,0.009}$	$\mathbf{0.970\,\pm\,0.015}$	$\mathbf{0.952\,\pm\,0.035}$	$\mathbf{0.966\,\pm\,0.015}$
Gradient Boost	$0.964\,\pm\,0.021$	$0.975\,\pm\,0.010$	$0.968\,\pm\,0.016$	$0.949\,\pm\,0.036$	$0.964\,\pm\,0.015$
Ridge Regression	$0.521\,\pm\,0.055$	$0.789\,\pm\,0.017$	$0.663\,\pm\,0.074$	$0.588\,\pm\,0.099$	$0.640\,\pm\,0.032$
Linear Regression	$0.521\,\pm\,0.055$	$0.789\,\pm\,0.017$	$0.663\,\pm\,0.074$	$0.588\,\pm\,0.099$	$0.640\,\pm\,0.032$
Neural Network	$0.951\,\pm\,0.026$	$0.965\,\pm\,0.011$	$0.943\,\pm\,0.019$	$0.908\,\pm\,0.038$	$0.942\,\pm\,0.016$

Appendix

Table 13: Multi-seed composite model-selection scores for reconstructing the one-vs-rest SVM decision functions in the four-class contrastive latent space. Each entry reports the mean

\pm

standard deviation of

Q_{\mathrm{composite}}

over seeds, computed from out-of-fold predictions of the corresponding reconstruction model. The composite score combines calibrated reconstruction

R^{2}

, Spearman rank correlation

\rho

, sign accuracy

\vartheta

, and the range-normalized MAE-complement score

S_{\mathrm{MAE}}

. Larger values indicate better reconstruction. Bold entries mark the best model for each boundary and the best average across boundaries.

Panel of Figure 8 — Figure 8: A splice diagram represented by a digraph. The weights on outgoing edges from nodes are depicted as $w_{v_{i}v_{j}}$ and the weights on outgoing edges from leaves are omitted.

Panel of Figure 9 — Figure 9: Starting point for algorithm

Panel of Figure 10 — Figure 10: The infinite linear graph used as a building block within the algorithm.

Panel of Figure 11 — Figure 11: Graph Refinement

Panel of Figure 25 — Algorithm 2 Generate maximal splice diagram from splice diagram

Figure 12: Heatmap of correlation matrix of the features used as input to regression networks and their binary labels.

Figure 13: Full Sequence. Mean confusion matrices across five seeds for all HOM and TOP experiments.

Figure 14: Full Sequence. Logit space visualization for a representative seed ( seed -456) for all HOM and TOP experiments. For display only, the two logits are standardized within each panel and central quantile filtering is used to avoid axis ranges being determined by a small number of extreme logits.

Figure 15: Full Sequence. Mean normalized feature importance scores for feature saliency, permutation importance, and SHAP; averaged over all five seeds for all HOM and TOP experiments.

Figure 16: Value-Index. Mean confusion matrices across five seeds for all HOM and TOP experiments.

Figure 17: Value-Index. Logit space visualization for a representative seed ( seed -42) for all HOM and TOP experiments.

Figure 18: Value-Index. Mean normalized feature importance scores for feature saliency, permutation importance, and SHAP; averaged over all five seeds for all HOM and TOP experiments.

Figure 19: Latent Space. Mean confusion matrices across five seeds for all HOM and TOP experiments.

Figure 20: Latent Space. Logit space visualization for a representative seed ( seed -42) for all HOM and TOP experiments.

Figure 21: Latent Space. Mean normalized feature importance scores for feature saliency, permutation importance, and SHAP; averaged over all five seeds for all HOM and TOP experiments.

Figure 22: Multi-class Latent Space; seed -42. Top Left: SVM decision boundaries in latent space separating STAR-Sph, STAR-Non, H-Sph, and H-Non classes. TOP Right: SVM prediction confidence map. Bottom Left: SVM support vectors. Bottom Right: SVM and linear, binary decision boundaries overlaid with seed -42 binary boundaries displayed as thick dashed lines and the linear boundaries for other seeds rendered as faint, dashed lines.

Panel of Figure 23 — Figure 23: Learning $SL_{2}(\mathbb{Z})$ orbit. (a) In-distribution $SL_{2}(\mathbb{Z})$ -orbit regression. Points show the ensemble-mean transformer prediction $\widehat{\alpha}$ against the true orbit size $\alpha$ over the unique held-out samples appearing across the $30$ -seed robustness run. Color indicates the standard deviation of the prediction across seeds. The dashed line is the ideal relation $\widehat{\alpha}=\alpha$ , while the solid line is the fitted calibration. The ensemble prediction shows strong in-distribution interpolation but visible seed sensitivity and degradation for some large-orbit examples. (b) Seed overlay where each point is a held-out prediction from one of the $30$ transformer seeds. The dashed line is the ideal relation $\widehat{\alpha}=\alpha$ . The overlay shows that most seeds learn the same global linear trend, but the spread increases for large orbit sizes and for a small set of lower- $\alpha$ outliers.

Figure 24: Learning $d$ from $\widehat{Z}$ . Normalized mean exponent feature importance.

Figure 25: Error profile for learning

d

exactly, on the

n=685

three-fiber dataset

\mathcal{D}_{3}^{20}

. Top left: exact-integer accuracy at each stage (from left: direct rounding with

\Delta

appended, the halved target

t/2

, all nineteen offsets, the symmetrized Dedekind features), mean

\pm

one standard deviation over the ten seeds; the offsets-only starting point (

17.9\%

) is discussed in the text. The remaining panels show out-of-fold predictions under the seed-42 fold assignment, for which the best model is exact on

89.2\%

(

611/685

), every miss off by exactly

\pm 2

t

. Top right: predicted vs. true

t=d+\Delta-\tfrac{1}{2}

. Bottom left: residual histograms of the best model (halved target, MAE

0.277

) and of the first bar’s rounding model (full target, MAE

0.950

) under the same folds; each is plotted in its own rounding variable, so

|\varepsilon|<\tfrac{1}{2}

marks a correctly rounded prediction in both, and the axis is truncated to

[-2,2]

. Bottom right: best-model

|\varepsilon|

against

\Delta

, with the misses marked.

Topological Representation Learning

Topological representation learning uses connectivity, holes, and higher-order structure to build or analyze machine-learning features.

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Topological Signatures of Grokking
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Topological Signatures of Grokking

Yifan Tang, Qiquan Wang, Inés García-Redondo, Anthea Monod

Imperial College London · Queen Mary University of London · University of Fribourg

Neural Network Generalization Neural Representation Geometry Grokking

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We study the grokking phenomenon through the lens of topology. Using persistent homology on point clouds derived from the embedding matrices of a range of models trained on modular arithmetic with varying primes, we identify a clear and consistent topological signature of grokking: a sharp increase in both the maximum and total persistence of first homology (

H_1

). Persistence diagrams reveal the emergence of a dominant long-lived topological feature together with increasingly structured secondary features, reflecting the underlying cyclic structure of the task. Compared to existing spectral and geometric diagnostics -- specifically, Fourier analysis and local intrinsic dimension -- persistent homology provides a unified geometric and topological characterization of representation learning, capturing both local and global multi-scale structure. Ablations across data regimes and control settings show that these topological transitions are tied to generalization rather than memorization. Our results suggest that persistent homology offers a principled and interpretable framework for analyzing how neural networks internalize latent structure during training.

Abstract

Topological Representation Learning

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