Knot So Simple: A Minimalistic Environment for Spatial Reasoning

TL;DR

KnotGym presents a minimal yet challenging visual knot-manipulation environment to study spatial reasoning over deformable ropes using only image observations. It defines a measurable complexity axis via the number of crossings $#X$ and Gauss codes $GC$, enabling a principled curriculum and generalization tests across train/test splits and increasing task difficulty. The paper benchmarks RL (PPO, DreamerV3, TD-MPC2) and prompting-based VLMs, finding that unknot is approachable while tie and convert present substantial generalization and data-efficiency challenges due to the need to decode and act upon topological goals. By highlighting acute perception, continuous spatial reasoning, and a very large search space, KnotGym provides a rigorous testbed for future agents—ranging from world-models to multi-modal reasoning—that fuse perception, planning, and grounded manipulation in deformable-object contexts.

Abstract

We propose KnotGym, an interactive environment for complex, spatial reasoning and manipulation. KnotGym includes goal-oriented rope manipulation tasks with varying levels of complexity, all requiring acting from pure image observations. Tasks are defined along a clear and quantifiable axis of complexity based on the number of knot crossings, creating a natural generalization test. KnotGym has a simple observation space, allowing for scalable development, yet it highlights core challenges in integrating acute perception, spatial reasoning, and grounded manipulation. We evaluate methods of different classes, including model-based RL, model-predictive control, and chain-of-thought reasoning, and illustrate the challenges KnotGym presents. KnotGym is available at https://github.com/lil-lab/knotgym.

Figures (13)

• Figure 1: KnotGym is a visual reasoning knot manipulation environment. It includes three tasks: transforming complex knots to a simple loop (example in the center); tying a loop into complex knots (right pane); and converting one knot into another, given a goal knot image. KnotGym has a continuous visual observation space (left pane) and an action space of applying forces to contact points (right pane), abstracting the specifics of robot end effectors. The space of goals, specified using Gauss code, is a factorial of the number of crossings (#X), which creates a ladder of generalization. Each goal defines an easily testable equivalence class over a continuous set of states.
• Figure 2: An episode is successful when the current knot configuration has the goal Gauss code. We obtain the Gauss code of any knot by traversing through the rope, starting from the white segment towards red (black arrow). When traversing, we denote an over-cross with +, and an under-cross with -, until we return to the starting segment.
• Figure 3: Train success rates of RL methods on nine different KnotGym setups after 1M environment steps during training. Error bars represent 95% confidence interval. All methods show non-trivial improvements on unknot via RL training, but struggle on tie and convert. No methods outperform a random policy at #X=4 of tie and convert, suggesting that increasing #X raises task difficulty significantly for tasks with many possible goals.
• Figure 4: Training curves for different number of goal configurations in the training set (DreamerV3, tie, #X=3).
• Figure 5: Generalization matrices for three tasks. Each entry of the matrices is success rate evaluated on the test split with $N$=128 episodes.