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.

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