Physical Knot Classification Beyond Accuracy: A Benchmark and Diagnostic Study

Shiheng Nie; Yunguang Yue

cs.CVMar 24, 2026

Physical Knot Classification Beyond Accuracy: A Benchmark and Diagnostic Study

Authors: Shiheng Nie, Yunguang Yue

Organizations: College of Science, Shihezi University, Shihezi 832003, Xinjiang, China

Abstract

Physical knot classification is a challenging fine-grained recognition task in which the intended discriminative cue is rope crossing structure; however, high closed-set accuracy may still arise from low-level appearance shortcuts rather than genuine topological understanding. In this work, we introduce dataset (1,440 images, 10 classes), which trains models on loosely tied knots and evaluates them on tightly dressed configurations to probe whether structure-guided training yields topology-specific gains. We demonstrate that topological distance successfully predicts residual inter-class confusion across multiple backbone architectures, validating the utility of our topology-aware evaluation framework. Furthermore, we propose topology-aware centroid alignment (TACA) and an auxiliary crossing-number prediction objective as two complementary forms of structural supervision. Notably, Swin-T with TACA achieves a consistent positive specificity gain (Delta_spec = +1.18 pp) across all random seeds under the canonical protocol, and auxiliary crossing-number prediction exhibits robust performance across data regimes without the real-versus-random reversal observed for centroid alignment. Causal probes reveal that background changes alone flip 17-32% of predictions and phone-photo accuracy drops by 58-69 percentage points, underscoring that appearance bias remains the principal obstacle to deployment. These results collectively demonstrate that our diagnostic workflow provides a principled and practical tool for evaluating whether a hand-crafted structural prior delivers genuine task-relevant benefit beyond generic regularization.

Figures & tables

Figure 1: Diagnostic workflow for prior-specific evaluation on Knots-10. Baseline training and a Mantel pre-check first decide whether a candidate structural prior is worth testing. If the Mantel signal is significant, the workflow compares real-prior and random-prior training through

\Delta_{\text{spec}}

and only then proceeds to causal probes and cross-domain tests.

ID	Knot Name	Type	$C_{\text{vis}}$	Family
OHK	Overhand Knot	Prime ( $3_{1}$ )	3	Stopper
SK	Slip Knot	Slip ( $3_{1}$ )	3	Stopper
F8K	Figure-8 Knot	Prime ( $4_{1}$ )	4	Stopper
BK	Bowline	Loop	4	Loop
F8L	Figure-8 Loop	Loop ( $4_{1}$ )	4	Loop
ABK	Alpine Butterfly	Loop	4	Loop

Table 1: Topological taxonomy of Knots-10.

C_{\text{vis}}

denotes the visual crossing number (see Remark below). These properties define the topological distance metric in Section 3.2 .

General Architectures
Model	Params	Test Acc (%)	Macro F1
ResNet-18	11.2M	$96.88\pm 1.06$	$0.969\pm 0.011$
ResNet-50	23.5M	$95.83\pm 1.03$	$0.959\pm 0.010$
EfficientNet-B0	4.0M	$96.25\pm 0.45$	$0.963\pm 0.005$
ViT-B/16	85.8M	$96.39\pm 0.35$	$0.964\pm 0.003$
Swin-T	27.5M	$\mathbf{97.22\pm 1.09}$	$\mathbf{0.972\pm 0.011}$

Table 2: Classification results on Knots-10 (mean

\pm

std across seeds 42, 123, and 456). All models use ImageNet pretraining and the tightness-stratified protocol.

Model	Spearman $\rho$	Pearson $r$	Mantel $r$	Mantel $p$
ResNet-18	$-0.291$	$-0.301$	$-0.291$	0.058
ResNet-50	$-0.472$	$-0.453$	$-0.472$	0.002 ∗∗
EfficientNet-B0	$-0.491$	$-0.426$	$-0.492$	0.001 ∗∗∗
ViT-B/16	$-0.398$	$-0.359$	$-0.398$	0.005 ∗∗
Swin-T	$-0.214$	$-0.195$	$-0.214$	0.173
${}^{}p<0.01$ ; ${}^{*}p<0.001$ . 9,999 permutations, two-tailed.

Table 3: Correlation between topological distance and pairwise confusion rate. Mantel

p

-values use 9,999 permutations to account for matrix non-independence ( 33 ) .

All values in %. Per-seed $\Delta_{\text{spec}}$ : Swin-T ( $+$ 1.05, $+$ 1.04, $+$ 1.46),
Condition	ResNet-50	Swin-T	ViT-B/16
CE only	$95.76\pm 0.71$	$97.15\pm 0.79$	$97.29\pm 0.17$
TACA (rand)	$96.46\pm 0.78$	$97.64\pm 0.35$	$95.56\pm 2.35$
TACA (real)	$96.18\pm 1.58$	$\mathbf{98.82\pm 0.26}$	$96.46\pm 1.06$
$\Delta_{\text{spec}}$ (pp)	$-0.28\pm 1.82$	$\mathbf{+1.18\pm 0.20}$	$+0.90\pm 1.97$
ViT-B/16 ( $+$ 3.13, $+$ 1.25, $-$ 1.67), ResNet-50 ( $-$ 2.71, $+$ 0.21, $+$ 1.67).

Table 4: TACA ablation under the canonical protocol (mean

\pm

std across seeds 42, 123, and 456;

n=480

\Delta_{\text{spec}}

= TACA(real)

-

TACA(rand). For protocol sensitivity, compare with the reduced-data suite in Table 5 .

Method	50% data	100% data
CE (baseline)	$94.31\pm 1.58$	$97.15\pm 0.96$
TACA (real)	$96.67\pm 0.63$	$96.67\pm 0.42$
TACA (rand)	$96.60\pm 0.43$	$98.19\pm 0.73$
AuxCrossing-Ord	$96.60\pm 0.97$	$97.08\pm 0.42$
$\Delta_{\text{spec}}$ (pp)	$+0.07$	$-1.53$

Table 5: Reduced-data stress test for Swin-T (mean

\pm

std across seeds 42, 123, and 456). This is an independent experimental suite with a different TACA normalization, so only within-table comparisons are valid.

\Delta_{\text{spec}}

= TACA(real)

-

TACA(rand).

Figure 2: Appearance bias under causal and deployment shifts. (A) Accuracy under original, rope-only, and background-swapped conditions for four representative models (

n=480

). (B) Prediction flip rate under background swapping, where Swin-T with TACA achieves the lowest flip rate among all evaluated models. (C) Accuracy under the phone-photo pilot across progressively shifted capture conditions (

n=100

), quantifying the deployment gap between controlled and real-world acquisition.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Configuration	Test Accuracy (%)	Macro F1
ResNet-18 (CE)	$96.88\pm 1.06$	$0.969\pm 0.011$
ResNet-18 (CE + TACA)	$96.81\pm 1.00$	$0.968\pm 0.010$
ResNet-18 (CE + TAML)	$96.53\pm 0.79$	$0.965\pm 0.008$
ResNet-18 (CE + TACA + TAML)	$95.97\pm 1.47$	$0.960\pm 0.014$
ResNet-50 (CE)	$95.83\pm 1.03$	$0.959\pm 0.010$
EfficientNet-B0 (CE)	$96.25\pm 0.45$	$0.963\pm 0.005$

Appendix

Table S1: Multi-seed robustness analysis. Each configuration is trained with seeds 42, 123, and 456 under the same protocol and evaluated on the tightness-stratified test set.

Loss Configuration	Test Acc (%)	Macro F1
CE only	$96.88\pm 1.06$	$0.969\pm 0.011$
CE + TACA	$96.81\pm 1.00$	$0.968\pm 0.010$
CE + TAML	$96.53\pm\mathbf{0.79}$	$0.965\pm 0.008$
CE + TACA + TAML	$95.97\pm 1.47$	$0.960\pm 0.014$

Appendix

Table S2: Loss-component ablation on ResNet-18 (mean

\pm

std across seeds 42, 123, and 456).

Photo	Factor Tested	Tightness	Angle	Background	Rope
1	Baseline	Tight	90°	Gray	A
2–3	Viewing angle	Tight	45°/30°	Gray	A
4–5	Looseness ( $\pm$ angle)	Loose	90°/45°	Gray	A
6–7	Background	Tight	90°	Blue/Beige	A
8	Lighting	Tight	90°	Gray	A †
9	Rope material	Tight	90°	Gray	B

Appendix

Table S3: Cross-domain photography protocol. Each knot type has 10 photographs with controlled changes relative to the training distribution. Photo 1 is the closest to the training setup.

Figure S1: Validation and test accuracy for all eight architectures, seed 42. Models are grouped into general-purpose and FGVC-specialized methods. See Table S1 for multi-seed results.

Figure S2: Per-class F1 scores for all eight architectures, seed 42. Darker green indicates higher F1. The horizontal line separates general-purpose and FGVC-specialized methods.

Figure S3: Training and validation accuracy curves for all eight architectures over 20 epochs. The top row shows general-purpose models and the bottom row FGVC-specialized methods.

Figure S4: Weight sensitivity analysis. Left: Spearman

\rho

across 200 random weight perturbations (

\pm 50\%

) for EfficientNet-B0. Right: default versus uniform weights across the five general-purpose architectures.

Figure S5: Grad-CAM visualizations for ResNet-18 (middle row) and ResNet-50 (bottom row) across all 10 knot classes. The top row shows the input images. Green labels mark correct predictions and red labels mark errors.

Figure S6: Distance matrices for ResNet-18. Left: topological distance. Center: baseline embedding distance. Right: TACA embedding distance. Darker colors indicate larger distances.

	OHK	SK	F8K	BK	F8L	ABK	CH	RK	FSK	FMB
OHK	—	0.10	0.50	0.50	0.50	0.50	0.50	0.50	0.50	0.50
SK		—	0.50	0.50	0.50	0.50	0.50	0.50	0.50	0.50
F8K			—	0.50	0.10	0.50	0.50	0.50	0.50	0.15
BK				—	0.50	0.50	0.50	0.50	0.50	0.50
F8L					—	0.50	0.50	0.50	0.50	0.50
ABK						—	0.50	0.50	0.50	0.50

Appendix

Table S4: Complete

\delta_{5}

(structural derivation) matrix for all 45 knot pairs. Four pairs receive reduced penalties; all others use 0.5. Together with Table 1 in the main text, this specifies the full

10\times 10

topological distance matrix.

Figure S7: Evolution of the five learnable topology weights for ResNet-18. All weights start at 0.2 and stabilize within about 10 epochs.

Figure S8: Mantel null distributions for the five general-purpose architectures (9,999 permutations each). Red dashed lines mark the observed Spearman correlations, and orange dotted lines mark the two-tailed

\alpha=0.05

thresholds.

Figure S9: McNemar test $p$ -values for within-group pairwise comparisons (seed 42, $n=480$ ). Color shows $-\log_{10}(p)$ . Diagonal cells report test accuracy. Cross-group comparisons are omitted because the test-set order differs.

Backbone	Configuration	Test Acc	Multi-seed Mean	Embed $\rho$
ResNet-18	CE only	95.83%	$96.88\pm 1.06$	0.462
	CE + TACA (hand-crafted)	97.08%	$96.81\pm 1.00$	0.645
	CE + TACA (learnable)	96.67%	$95.28\pm 2.23$	0.509
ResNet-50	CE only	96.04%	$95.83\pm 1.03$	0.495
	CE + TACA (hand-crafted)	95.83%	— †	0.641
	CE + TACA (learnable)	96.67%	$96.53\pm 0.43$	0.505

Appendix

Table S5: Hand-crafted versus learnable topology weights for ResNet-18 and ResNet-50. Test accuracy uses seed 42; multi-seed results are shown where available.

Dataset	Method	Test Acc	Macro F1	Embed $\rho$
Knots-10	CE only	96.04%	0.961	0.495
Knots-10	CE + TACA	95.83%	0.959	0.641
CUB-200-2011	CE only	81.81%	0.819	0.109
CUB-200-2011	CE + TACA	80.67%	0.807	0.086

Appendix

Table S6: Cross-dataset generalization of TACA. ResNet-50 is evaluated on Knots-10 and CUB-200-2011 with dataset-specific distance matrices. For Knots-10, the seed-42 rows are shown for consistency with the cross-dataset comparison; the main paper reports multi-seed results.

Swin-T variants (within-architecture comparison)
Model	Attribution rope ratio (%)
Swin-T CE	$8.87\pm 3.24$
Swin-T TACA	$7.96\pm 3.48$
Swin-T AuxCrossing	$7.72\pm 3.46$
CNN baseline
ResNet-50 CE	$38.30\pm 12.06$

Appendix

Table S7: Grad-CAM attribution rope ratio (mean

\pm

std,

n=100

). The metric is the fraction of attribution mass inside the rope mask. Only within-architecture comparisons are valid.

	50% data			100% data
Method	s42	s123	s456	s42	s123	s456
CE	95.00	92.50	95.42	96.04	97.71	97.71
TACA (real)	97.29	96.67	96.04	97.08	96.67	96.25
TACA (rand)	96.25	97.08	96.46	98.96	98.13	97.50
AuxCrossing-Ord	95.83	97.71	96.25	97.50	96.67	97.08

Appendix

Table S8: Per-seed results for the reduced-data stress test on Swin-T. All values are test accuracy (%) on

n=480

images.

Classification

Classification maps observations to discrete labels, including familiar, unknown, hierarchical, or multiple categories.

See all 408 papers

Feb 19, 2026

Shortcut learning in geometric knot classification

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.

Djordje Mihajlovic, Davide Michieletto

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 +1

Shortcut Learning

Ask PDF

Jul 22, 2026

Writhe-Based Polymer Link Classification Using Machine Learning

Unique and rapid classification of knots and links is an open mathematical problem that is relevant to a range of (bio)physical systems, including polymer melts, DNA, and proteins. In this paper, we explore a data-driven approach to the classification problem of link topology. Extending the framework introduced in Ref. 1 (Sleiman et al, 2024 Soft Matter, 20(1), pp.71-78), we show that a feedforward neural network trained on the writhe density matrix classifies thermally equilibrated configurations of the first six prime links with 97% accuracy. We demonstrate that this accuracy remains high across a range of temperatures and lengths of link components, while rapidly deteriorating with the addition of topology-altering Gaussian noise; a result consistent with the writhe density matrix containing features sensitive to topology. Our results show that neural networks based on the writhe density matrix efficiently classify two-component links, establishing machine learning as a promising tool for rapid classification of more complex link topologies, e.g. Borromean rings and multi-component links, as the computational cost of exact numerical calculation of topological invariants becomes prohibitive.

Jack Beda, Djordje Mihajlovic, Kasturi Barkataki +1

School of Physics and Astronomy, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK. · School of Mathematics, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK. · MRC Human Genetics Unit, Institute of Genetics and Cancer, University of Edinburgh, Edinburgh EH4 2XU, UK

Topic pending

Ask PDF

Mar 13, 2026

Topo-R1: Detecting Topological Anomalies via Vision-Language Models

Topology is critical in tubular structures such as blood vessels, nerve fibers, and road networks, where connectivity and loop structure govern downstream functional analysis. Vision-Language Models (VLMs) are promising candidates for understanding such structures, given their reasoning and grounding capabilities. To probe their topological perception, we systematically evaluate leading closed- and open-source VLMs on localizing and classifying four canonical topological anomalies (broken/spurious connections, missing/extra branches) in tubular-network segmentation masks. They perform nearly at random, indicating that topology-aware perception is largely absent from current general-purpose VLMs. As no existing resource pairs segmentation masks with localized anomaly annotations, we build an automated, multi-domain data-curation pipeline that synthesizes diverse topological perturbations with verifiable Betti-number annotations across graduated difficulty levels, yielding the first systematic benchmark with a large-scale training set and held-out in-distribution (ID) and out-of-distribution (OOD) test suites. Building on this benchmark, we introduce Topo-R1, centered on a topology-aware composite reward that jointly scores localization, classification, and skeleton-level structural fidelity. Supervised fine-tuning cold-starts schema-compliant outputs, and Group Relative Policy Optimization (GRPO) then optimizes the policy against this reward, steering predictions toward topologically meaningful structure rather than superficial pixel overlap. Extensive experiments show that Topo-R1 substantially outperforms general-purpose VLMs and matches or exceeds supervised baselines across ID, OOD, and real-segmentation-output protocols, establishing a strong foundation for VLM-based topological understanding of structured visual data.

Meilong Xu, Qingqiao Hu, Xiaoling Hu +6

Stony Brook University · Massachusetts General Hospital and Harvard Medical School · Stanford University +1

Anomaly Detection VLM Evaluation

Ask PDF

Abstract

Classification

Explore similar work

Shortcut learning in geometric knot classification

Writhe-Based Polymer Link Classification Using Machine Learning

Topo-R1: Detecting Topological Anomalies via Vision-Language Models