# CurveBench: A Benchmark for Exact Topological Reasoning over Nested Jordan Curves

arXiv: https://arxiv.org/abs/2605.14068 · HTML: https://mlsanity.com/paper/2605.14068/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Multimodal reasoning, one of 17 directions in LLM reasoning.
- Builds on 21 of its 28 references that are in the map.
- First posted on arXiv 13 May 2026.

## Placed in

- LLM reasoning / [Multimodal reasoning](https://mlsanity.com/lr/multimodal-reasoning/)

## Builds on (most cited first, 21 of 21)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [End-to-End Object Detection with Transformers](https://mlsanity.com/paper/2005.12872/) (2020)
- [Qwen3-VL Technical Report](https://mlsanity.com/paper/2511.21631/) (2025)
- [Visual Genome: Connecting Language and Vision Using Crowdsourced Dense Image Annotations](https://mlsanity.com/paper/1602.07332/) (2016)
- [CLEVR: A Diagnostic Dataset for Compositional Language and Elementary Visual Reasoning](https://mlsanity.com/paper/1612.06890/) (2016)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Tulu 3: Pushing Frontiers in Open Language Model Post-Training](https://mlsanity.com/paper/2411.15124/) (2024)
- [A Diagram Is Worth A Dozen Images](https://mlsanity.com/paper/1603.07396/) (2016)
- [VLM-R1: A Stable and Generalizable R1-style Large Vision-Language Model](https://mlsanity.com/paper/2504.07615/) (2025)
- [Neural Motifs: Scene Graph Parsing with Global Context](https://mlsanity.com/paper/1711.06640/) (2017)
- [Pix2Struct: Screenshot Parsing as Pretraining for Visual Language Understanding](https://mlsanity.com/paper/2210.03347/) (2022)
- [Pix2seq: A Language Modeling Framework for Object Detection](https://mlsanity.com/paper/2109.10852/) (2021)
- [R1-VL: Learning to Reason with Multimodal Large Language Models via Step-wise Group Relative Policy Optimization](https://mlsanity.com/paper/2503.12937/) (2025)
- [Visual Spatial Reasoning](https://mlsanity.com/paper/2205.00363/) (2022)
- [IconQA: A New Benchmark for Abstract Diagram Understanding and Visual Language Reasoning](https://mlsanity.com/paper/2110.13214/) (2021)
- [LMM-R1: Empowering 3B LMMs with Strong Reasoning Abilities Through Two-Stage Rule-Based RL](https://mlsanity.com/paper/2503.07536/) (2025)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [Perception-R1: Pioneering Perception Policy with Reinforcement Learning](https://mlsanity.com/paper/2504.07954/) (2025)
- [RoadTracer: Automatic Extraction of Road Networks from Aerial Images](https://mlsanity.com/paper/1802.03680/) (2018)
- [Topological Map Extraction from Overhead Images](https://mlsanity.com/paper/1812.01497/) (2018)

## Built on by (most cited first, 1 of 1)

- [MindTopo: Can Foundation Models Reason in Topological Space?](https://mlsanity.com/paper/2609.11900/) (2026)

## Abstract (arXiv)

We introduce CurveBench, a benchmark for hierarchical topological reasoning from visual input. CurveBench consists of 756 images of pairwise non-intersecting Jordan curves across easy, polygonal, topographic-inspired, maze-like, and dense counting configurations. Each image is annotated with a rooted tree encoding the containment relations between planar regions. We formulate the task as structured prediction: given an image, a model must recover the full rooted containment tree induced by the curves. Despite the visual simplicity of the task, the strongest evaluated model, Gemini 3.1 Pro, achieves only 71.1% tree-generation accuracy on CurveBench-Easy and 19.1% on CurveBench-Hard. We further demonstrate benchmark utility through RLVR-style fine-tuning of open-weight vision-language models. Our trained Qwen3-VL-8B model improves over textttQwen-3-VL-8B-Thinking from 2.8% to 33.3% tree-generation accuracy on CurveBench-Easy, exceeding GPT-5.4 and Claude Opus 4.5 under our evaluation protocol. The remaining gap, especially on CurveBench-Hard, shows that exact topology-aware visual reasoning remains far from solved.