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Knot So Simple: A Minimalistic Environment for Spatial Reasoning
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Introduces KnotGym, a MuJoCo-based visual rope-knot manipulation environment with three Gauss-code-goal tasks of tunable crossing-number complexity, and benchmarks RL (PPO, DreamerV3, TD-MPC2) and VLM prompting (GPT-4.1-nano).
Problem
Existing spatial reasoning and manipulation benchmarks are static or use dense distance rewards; there is no visual, interactive environment where goals are topological equivalence classes with a quantifiable complexity axis for testing generalization.
Approach
Defines knot manipulation goals via Gauss codes (topological equivalence classes), so success is judged by crossing structure rather than coordinate distance, with the number of crossings providing a natural complexity-generalization ladder from pure image observations.
Key results
- DreamerV3 reaches 93.3% train success on unknot with 2 crossings after 1M steps, versus 11.1% for random.
- On tie with 3 crossings, DreamerV3 reaches 16.1% vs 2.5% random; at 4 crossings no RL method beats random on tie or convert.
- GPT-4.1-nano prompting achieves 0.0% in open mode on all tasks and at most 20.9% (stateful, unknot 2 crossings).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Train DreamerV3 rather than use a random policy | KnotGym unknot task · unknot, 2–4 crossings (train) | success rate: 11.1 -> 93.3; 8.4 -> 93.4; 6.7 -> 89.3 | Helped |
| Train DreamerV3 rather than use a random policy | KnotGym tie task · tie, 2–4 crossings (train) | success rate: 35.9 -> 83.2; 2.5 -> 16.1; 1.4 -> 4.1 | Helped |
| Train DreamerV3 rather than use a random policy | KnotGym convert task · convert, 2–4 crossings (train) | success rate: 36.8 -> 71.5; 9.2 -> 15.3; 2.9 -> 5.4 | Helped |
| Increase knot crossings from 2 to 4 | KnotGym tie and convert tasks · tie and convert (train) | success rate: tie DreamerV3: 83.2 -> 4.1; convert DreamerV3: 71.5 -> 5.4 | Hurt |
| Use stateful prompting with observation history instead of stateless prompting | GPT-4.1-nano · unknot, 2–4 crossings (train) | success rate: 12.0 -> 20.9; 6.7 -> 17.0; 6.1 -> 7.4 | Helped |
| Use stateful instead of stateless prompting | GPT-4.1-nano · tie and convert, 2–4 crossings (train) | success rate: tie: 7.6 -> 5.5, 1.0 -> 0.9, 0.0 -> 0.0; convert: 5.7 -> 2.8, 1.0 -> 0.9, 0.0 -> 0.0 | Mixed |
| Increase TD-MPC2 policy size from the default 5M to 48M parameters (ablation) | TD-MPC2 on KnotGym · KnotGym | success rate | No clear effect |
| Require maintaining the goal Gauss code for multiple frames instead of terminating on first success (ablation) | KnotGym reward and termination · KnotGym tasks | task difficulty | Hurt |
Surprising findings
- Prompted GPT-4.1-nano performs worse than a random policy on most task settings, except unknot with 2-3 crossings.
- Constraining goal diversity during training did not trivially resolve tie difficulty; generalization to new same-complexity goals stays weak for tie.
Limitations
- Simulator throughput limited to CPUs (about 20 steps/s per CPU), with scaling to more crossings requiring fidelity and resolution tuning.
- RL methods show poor generalization to new goals and to higher crossing numbers on tie and convert.
- Prompted VLMs cannot produce precise grounded actions; only one proprietary small VLM was evaluated.
Experimental setup
Datasets: simulated knot configurations generated by MuJoCo random policy with manual filtering
Benchmarks: KnotGym unknot KnotGym tie KnotGym convert
Metrics: success rate
Baselines: PPO DreamerV3 TD-MPC2 GPT-4.1-nano random policy
Builds on: PPO DreamerV3 TD-MPC2 Gymnasium MuJoCo
Closest papers by meaning
- Localizing, Structuring, and Rendering: Bridging 3D and 2D Vision-Language-Action Models for Robotic Manipulation (CVPR 2026 · similarity 0.87)
- SpatialClaw: Rethinking Action Interface for Agentic Spatial Reasoning (NeurIPS 2026 · similarity 0.85)
- Learning Spatial-Aware Manipulation Ordering (NeurIPS 2025 · similarity 0.85)
- Generate Subgoal Images before Act: Unlocking the Chain-of-Thought Reasoning in Diffusion Model for Robot Manipulation with Multimodal Prompts (CVPR 2024 · similarity 0.85)
- OmniManip: Towards General Robotic Manipulation via Object-Centric Interaction Primitives as Spatial Constraints (CVPR 2025 · similarity 0.85)
- Unfolding Spatial Cognition: Evaluating Multimodal Models on Visual Simulations (ICLR 2026 · similarity 0.85)
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