{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2021-12-01-deepmind-knot-theory-intuition/","md":"https://postcutoff.com/e/2021-12-01-deepmind-knot-theory-intuition/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2021-12-01-deepmind-knot-theory-intuition","date":"2021-12-01","date_precision":"day","short_title":"DeepMind and mathematicians use machine learning to guide new theorems in knot theory and representation theory","deck":null,"takeaway":"Davies et al. (Nature, Dec 2021) used supervised learning plus attribution to point mathematicians to hidden relationships.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"Advancing mathematics by guiding human intuition with AI (Nature)","url":"https://www.nature.com/articles/s41586-021-04086-x","type":"paper","group":"primary","domain":"nature.com"},{"n":2,"title":"Critical review of the paper (arXiv 2112.04324)","url":"https://arxiv.org/abs/2112.04324","type":"discussion","group":"reactions","domain":"arxiv.org"}],"official":1,"filed":"2026-09-29","updated":"2026-09-29","orgs":["DeepMind","University of Oxford","University of Sydney"],"title":"DeepMind and mathematicians use machine learning to guide new theorems in knot theory and representation theory","summary":"Davies et al. (Nature, Dec 2021) used supervised learning plus attribution to point mathematicians to hidden relationships. That led to a new theorem linking the knot signature to hyperbolic geometry, and to progress on the combinatorial invariance conjecture for Kazhdan–Lusztig polynomials.","key_facts":["Nature 600:70–74 (2021)","Knot theory: new relation between signature and the 'natural slope' (Lackenby, Juhász); follow-up in Geometry & Topology (2024)","Representation theory: progress towards the combinatorial invariance conjecture (Williamson)","Humans stated and proved the theorems; ML highlighted which features mattered"],"key_numbers":[],"tags":["math","knot-theory","representation-theory","nature"],"science":{"field":"mathematics","subfield":"knot theory / representation theory","problem":"Combinatorial invariance conjecture for Kazhdan–Lusztig polynomials; relations between knot invariants","result":"ML-guided discovery of a conjectured, then proved, relation between the knot signature and hyperbolic invariants, and a new approach to combinatorial invariance for symmetric groups.","open_since":"","ai_system":["supervised neural networks with gradient saliency"],"human_role":"Human-led with AI tools: ML suggested patterns; mathematicians formulated and proved the theorems","verification":{"key":"peer-review","label":"Peer-reviewed","text":"Peer-reviewed in Nature; human proofs"},"status":"confirmed","shock":""},"body_md":"## What happened\n\nDeepMind trained models to predict one mathematical quantity from others, then used attribution to show which inputs mattered, prompting expert mathematicians to formulate and prove new results.\n\n## Why it matters\n\nIt was the first Nature-level demonstration of AI contributing to pure maths research, as an intuition aid rather than a prover.","disputed":[],"related":[{"id":"2023-12-14-funsearch-cap-sets","url":"https://postcutoff.com/e/2023-12-14-funsearch-cap-sets/","date":"2023-12-14","date_precision":"day","short_title":"FunSearch: an LLM finds new cap-set constructions, the first LLM discovery in open maths","deck":null,"takeaway":"It was the direct ancestor of AlphaEvolve and showed that LLMs' mistakes don't matter when outputs can be automatically checked.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Google DeepMind","University of Wisconsin–Madison"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-09-29","type":"filed","text":"Created"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-29","run":null,"sources_read":null,"updated":"2026-09-29","human_review":null,"version":{"date":"2026-10-04"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":null,"in_training_data":true},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":null,"in_training_data":true},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":null,"in_training_data":true},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":null,"in_training_data":true}],"short_url":null}