[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-133841-en":3,"doc-seo-133841-105":31,"detail-sidebar-cat-0-en-105":92},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},133841,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Cactus doodles","Cactus doodles are combinatorial and geometric objects connected to cactus groups in the same structural way that knots relate to braids. The work defines cactus doodles using local moves on plane curves, proves they can be generated from cactus group elements via a closure procedure, and derives fundamental properties. The study also compares their behavior with classical doodles under mirror reflection and establishes key equivalence results for cactus doodle diagrams.","Bol. Soc. Mat. Mex. (2025) 31:3  \n[https://doi.org/10.1007/s40590-024-00681-w](https://doi.org/10.1007/s40590-024-00681-w)  \nORIGINAL ARTICLE  \nCactus doodles  \nJacob Mostovoy1 · Andrea Rincón-Prat1  \nReceived: 30 December 2023 / Accepted: 11 October 2024 © The Author(s) 2024  \nAbstract  \nCactus doodles are combinatorial/geometric objects that are related to cactus groups in the same way as knots are related to braids. We define them in terms of local moveson plane curves, show that they can be obtained from elements of the cactus group by a “closing” procedure and establish some of their basic properties.  \n1 Introduction and statement of results  \n1.1 What this note is about  \nThere exist many groups whose elements can be represented by collections ofdescending curves in a plane or in space and in this respect are similar to the braid group. Examples of such generalized braid groups include virtual braid groups [7], groups of welded braids [5], twin (or planar braid) groups [8] . In the same way as braidscan be “closed” so as to produce knots and links, other braid-like groups give rise to knot-like objects by means of the closure operation: virtual braids produce virtual links, welded braids produce welded links, and planar braids give rise to doodles (in the sense of Khovanov [8] rather than Fenn and Taylor [4]) . Many properties of these knot-like objects can be inferred from the properties of the corresponding braid-like groups although it is also true that some questions that are easy to answer for braids are hard (or still open) for links.  \nIn the present note, we introduce a new kind of knot-like objects that we call cactus doodles. These are curves on a sphere that may have self-intersections, considered up to a number of moves that mimic the relations in the cactus group. These moves generalize the moves on usual doodles.  \nWe also establish some basic properties of the cactus doodles. Namely, we show that every cactus doodle arises as a closure of an element of a cactus group and prove that any two equivalent cactus doodles whose diagrams cannot be simplified  \nB Jacob Mostovoy [jacob@math.cinvestav.mx](jacob@math.cinvestav.mx)  \nAndrea Rincón-Prat  \n[aprincon@math.cinvestav.mx](aprincon@math.cinvestav.mx)  \n1 Departamento de Matemáticas, CINVESTAV, Col. San Pedro Zacatenco, C.P. 07360 Mexico, D.F., Mexico  \nFig. 1 A “braid” representing s4 ,8  \nFig. 2 The relations in the cactus group  \ncan be obtained from each other by a sequence of moves that preserve the number of intersection points, and, possibly, a mirror reflection. The latter property is similar to the behavior of the usual doodles and stands in contrast to the case of knots and links. As a corollary, we see that the usual doodles, considered up to mirror reflections, area subset of the cactus doodles.  \n1.2 Cactus groups  \nFor n > 0, the cactus group Jn has the generators sp ,q , where 1 ≤ p \u003C q ≤ n, and the following relations:  \ns2p ,q = 1 ,  \nsp ,q sm ,r = sm ,rsp ,q if [ p , q]∩[m , r] = ∅ ,  \nsp ,q sm ,r = sp+q−r ,p+q−msp ,q if [m , r] ⊂ [ p , q] .  \nThere is a homomorphism of the cactus group Jn onto the symmetric group Sn: itsends sp ,q into the permutation of the ordered set 1 \u003C ··· \u003C n which reverses the order of p , p + 1 , . . . , q and leaves the rest of the elements unchanged. The kernel of this homomorphism is the fundamental group of the moduli space of stable real rational curves with n + 1 marked points; see, for instance Ref. [3] .  \nElements of Jn can be represented by “planar braids with self-intersections”; see Ref. [10] . The braid representing sp ,q has one point, where p − q + 1 strands meet, see Fig. 1. Then the product in Jn is simply the concatenation of braids. The relations have the form shown in Fig. 2.  \nIn these terms, the homomorphism Jn → Sn can be described as follows: the permutation defined by a braid is obtained by following its strands. In particular, sp ,q is sent to the permutation of the ordered set 1 \u003C ··· \u003C n which","cbCaib49tQazvDbj","https://ap.wps.com/l/cbCaib49tQazvDbj","pdf",1682574,2,1,16,"English","en",105,"# Introduction and statement of results\n## What this note is about\n## Cactus groups\n## Doodles and cactus doodles","[{\"question\":\"What are cactus doodles, according to the document?\",\"answer\":\"Cactus doodles are immersed closed curves on the sphere, considered up to smooth isotopy and specific elementary moves. Their defining condition requires that tangent lines at each k-tuple point are all distinct.\"},{\"question\":\"How are cactus doodles related to cactus groups?\",\"answer\":\"Every cactus doodle arises as the closure of an element of a cactus group. The construction uses local moves on curves that mirror relations in the cactus group.\"},{\"question\":\"What elementary moves are used to define equivalence of cactus doodles?\",\"answer\":\"Equivalence is generated by two types of elementary moves: one type generalizes relations that create or annihilate pairs of k-tuple intersection points, and another type passes a k-tuple intersection point through an n-tuple intersection point.\"}]","Cactus doodles | PDF",1787227694,40,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":29},"cactus-doodles","",{"@graph":37,"@context":86},[38,54,69],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,48,51],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":20},"https://docshare.wps.com/document/","Document",{"item":49,"name":12,"@type":44,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":44,"position":53},"https://docshare.wps.com/document/cactus-doodles/133841/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":42,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-31","2026-08-20",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What are cactus doodles, according to the document?","Question",{"text":76,"@type":77},"Cactus doodles are immersed closed curves on the sphere, considered up to smooth isotopy and specific elementary moves. Their defining condition requires that tangent lines at each k-tuple point are all distinct.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How are cactus doodles related to cactus groups?",{"text":81,"@type":77},"Every cactus doodle arises as the closure of an element of a cactus group. The construction uses local moves on curves that mirror relations in the cactus group.",{"name":83,"@type":74,"acceptedAnswer":84},"What elementary moves are used to define equivalence of cactus doodles?",{"text":85,"@type":77},"Equivalence is generated by two types of elementary moves: one type generalizes relations that create or annihilate pairs of k-tuple intersection points, and another type passes a k-tuple intersection point through an n-tuple intersection point.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":47,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":30,"slug":119},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":47,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":47,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":47,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":47,"category_name":137,"show_sort_weight":107,"slug":138},19,"General","general"]