[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-118152-en":3,"doc-seo-118152-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},118152,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Machine Learning Discovers Invariants of Braids and Flat Braids","Uses supervised machine learning to classify braid examples (including flat braids) as trivial or non-trivial, leveraging multilayer perceptron neural networks. When the classifiers achieve strong performance, their internal structure is interpreted to generate explicit mathematical conjectures. Those conjectures are then proved rigorously as theorems, yielding newly discovered invariants of braids. The study is motivated by earlier experiments on AI untangling braids, especially for braids with 3 strands.","Adv. Appl. Clifford Algebras (2024) 34:49 􀀂c The Author(s) 2024  \n[https://doi.org/10.1007/s00006-024-01349-4](https://doi.org/10.1007/s00006-024-01349-4)  \nMachine Learning Discovers Invariants of Braids and Flat Braids  \nAlexei Lisitsa, Mateo Salles and Alexei Vernitski∗  \nAbstract. We use machine learning to classify examples of braids (or flat braids) as trivial or non-trivial. Our machine learning takes the form of supervised learning, specifically multilayer perceptron neural networks. When they achieve good results in classification, we are able to interpret their structure as mathematical conjectures and then prove these conjectures as theorems. As a result, we find new invariants of braids and prove several theorems related to them. This work evolves from our experiments exploring how different types of AI cope with untangling braids with 3 strands, this is why we concentrate mostly on braids with 3 strands.  \n1. Introduction  \nThe authors’ recent research is in a type of artificial intelligence (AI) known as explainable AI; as applied to detecting trivial braids, which is the main topic of this article, this means that we used AI to produce a specific sequence of Reidemeister moves that untangles a braid [13, 14] . We also applied explainable AI to knots and used more abstract algebraic constructions, instead of Reidemeister moves, to detect trivial knots, with AI producing a proof demonstrating, in all detail, that a knot is trivial [5, 6 , 15] . Related recent research by other authors is [7, 12] . In this study we use a different approach; let us describe it from the point of view of machine learning (ML), on the one hand, and from the point of view of mathematics, on the other hand.  \nFrom the ML perspective, interpretability of ML models is an important area of research. Our study is a positive example in this direction. We curate our ML models, keeping them small and simple; this enables us to manually inspect details of neural networks in our ML models; hence, we manage to  \nThis article is part of the Topical Collection on Machine-Learning Mathematical Structures edited by Yang-Hui He, Pierre Dechant, Alexander Kasprzyk, and Andre Lukas.  \n∗ Corresponding author.  \nnotice some patterns in the trained ML models; based on them, we formulate mathematical conjectures and then prove them as theorems.  \nFrom the mathematical perspective, automated discovery of theorems or, in other words, AI-assisted conjectures is an increasingly important direction of research in recent years, see, for example, [3, 4 , 19] . In this paper we describe how our experiments with applying neural networks to braids and flat braids have enabled us to form conjectures, which we then were able to prove as Theorems 1 , 2 , 3. The results of this study are an inspiration to use supervised learning to formulate new conjectures, as part of mathematical research.  \n2. Braids: Definitions and Encodings  \nBraids have connections with a number of other mathematical constructions, including knots, groups [10] and Clifford algebras [11] .  \nThis paper concentrates on braids with 3 strands (unless stated otherwise); an example of such a braid is shown in Fig. 1. We visualize a braid as stretched from the left to the right. Denote positions of strands in the braid by 1 , 2 , 3 from the top to the bottom. Recall that the clockwise half-turn swapping the positions of two adjacent strands in positions i, i + 1 in a braid is denoted by σi [10, Section 1 .2.4]; thus, the braid in Fig. 1 is σ 1 σ2 σ −11 σ−21 . Denote the number of crossings in a braid or, equivalently, the number of generators σi featuring in the word describing the braid, by k; for example, for the braid in Fig. 1 we have k = 4 . In this study we use neural networks to work with braids, therefore, it is convenient to encode braids as matrices. We consider 4 encodings of braids as matrices, denoted below by EP2, EP1, ES2, ES1 . In each of the encodings, the braid is represented by a matrix containing ","cbCaik9ndz8CVgOv","https://ap.wps.com/l/cbCaik9ndz8CVgOv","pdf",758172,1,26,"English","en",105,"# Introduction\n## Braids: Definitions and Encodings","[{\"question\":\"What problem does the paper address using machine learning?\",\"answer\":\"The paper uses supervised machine learning to classify examples of braids (and flat braids) as trivial or non-trivial.\"},{\"question\":\"What type of neural network model is used in the study?\",\"answer\":\"It uses multilayer perceptron neural networks for the supervised classification task.\"},{\"question\":\"How do the authors turn model behavior into mathematical results?\",\"answer\":\"After obtaining good classification performance, the authors interpret the learned structure to form mathematical conjectures, then prove these conjectures as theorems.\"}]","Machine Learning Discovers Invariants of Braids and Flat Braids | PDF",1785681918,66,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"machine-learning-discovers-invariants-of-braids-and-flat-braids","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/machine-learning-discovers-invariants-of-braids-and-flat-braids/118152/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address using machine learning?","Question",{"text":75,"@type":76},"The paper uses supervised machine learning to classify examples of braids (and flat braids) as trivial or non-trivial.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What type of neural network model is used in the study?",{"text":80,"@type":76},"It uses multilayer perceptron neural networks for the supervised classification task.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the authors turn model behavior into mathematical results?",{"text":84,"@type":76},"After obtaining good classification performance, the authors interpret the learned structure to form mathematical conjectures, then prove these conjectures as theorems.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]