{"id":1022,"date":"2015-06-03T15:27:34","date_gmt":"2015-06-03T15:27:34","guid":{"rendered":"http:\/\/pageperso.math.univ-toulouse.fr\/hippocampe\/?p=1022"},"modified":"2015-06-10T14:08:22","modified_gmt":"2015-06-10T14:08:22","slug":"lycee-saint-sernin-toulouse-tresses-et-entrelacs","status":"publish","type":"post","link":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/","title":{"rendered":"Lyc\u00e9e Saint-Sernin (Toulouse) &#8211; Tresses et entrelacs"},"content":{"rendered":"<p style=\"text-align: justify\">Lundi, 9h : les \u00e9l\u00e8ves du lyc\u00e9e Saint-Sernin arrivent \u00e0 l&#8217;universit\u00e9 Paul Sabatier de Toulouse. Leur professeur de math\u00e9matiques les emm\u00e8ne \u00e0 l&#8217;institut de math\u00e9matiques o\u00f9\u00a0un enseignant-chercheur et trois doctorants sont l\u00e0 pour les accueillir. Le sujet sur lequel les \u00e9l\u00e8ves vont se casser la t\u00eate pendant trois jours est d\u00e9voil\u00e9 : ce sera les tresses et les entrelacs.<\/p>\n<p style=\"text-align: justify\"><a href=\"http:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2730.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1023 alignleft\" alt=\"IMG_2730\" src=\"http:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2730-300x225.jpg\" width=\"300\" height=\"225\" srcset=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2730-300x225.jpg 300w, https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2730-1024x768.jpg 1024w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><span style=\"text-align: justify\">Un sac de noeuds est l\u00e0, il ne reste plus qu&#8217;\u00e0 d\u00e9m\u00ealer les fils et \u00e0 les r\u00e9-emm\u00ealer. Et ensuite, que peut-on faire avec toutes ces ficelles ? On d\u00e9finit la somme de deux noeuds en les mettant l&#8217;un\u00a0\u00e0 la suite de l&#8217;autre. De la m\u00eame fa\u00e7on, on obtient le produit de deux tresses. Est-ce que la somme et le produit se comportent de la m\u00eame fa\u00e7on que la somme et le produit des entiers pour lesquels on a :<\/span><\/p>\n<ul>\n<li style=\"text-align: justify\">a + b = b + a et a . b = b . a, quels que soient les entiers a et b,<\/li>\n<li style=\"text-align: justify\">(a + b) + c = a + (b + c) et (a . b) . c = a . (b . c), quels que soient les entiers a, b et c ?<\/li>\n<\/ul>\n<div style=\"width: 310px\" class=\"wp-caption alignright\"><a href=\"http:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2729.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\" \" title=\"Noeud de tr\u00e8fle et une tresse\" alt=\"IMG_2729\" src=\"http:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2729-300x225.jpg\" width=\"300\" height=\"225\" \/><\/a><p class=\"wp-caption-text\">Le noeud de tr\u00e8fle et une tresse<\/p><\/div>\n<p style=\"text-align: justify\">Lorsque l&#8217;on dessine un noeud ou une tresse sur un tableau, on fait un choix de projection.\u00a0Plusieurs dessins peuvent provenir du m\u00eame noeud ou de la m\u00eame tresse. Essayez par exemple de dessiner la tresse \u00e0 droite avec uniquement un croisement.<\/p>\n<p style=\"text-align: justify\">\u00c0 l&#8217;aide de ces dessins, appel\u00e9s diagrammes de noeud, on cherche \u00e0 voir si deux noeuds sont diff\u00e9rents. \u00cates-vous vraiment certains que le noeud de tr\u00e8fle ne peut pas se d\u00e9m\u00ealer ?<\/p>\n<p style=\"text-align: justify\">Trois jours plus tard, les \u00e9l\u00e8ves de Saint-Sernin ont encore plein de questions mais aussi beaucoup de r\u00e9ponses. Ils les ont expliqu\u00e9es dans des posters qu&#8217;ils ont\u00a0pr\u00e9sent\u00e9s aux membres de l&#8217;institut de math\u00e9matiques. Les th\u00e8mes \u00e9tudi\u00e9s \u00e9taient les suivants :<\/p>\n<ul>\n<li>somme de deux noeuds,<\/li>\n<li>produit de deux tresses,<\/li>\n<li>mouvements de Reidemeister,<\/li>\n<li>invariants de noeuds et tresses,<\/li>\n<li>l&#8217;impl\u00e9mentation des noeuds dans un ordinateur et la recherche d&#8217;algorithme de comparaison de noeuds,<\/li>\n<li>nombre d&#8217;entrelacements et<\/li>\n<li>un probl\u00e8me de miroirs (que fait ce th\u00e8me ici ? la r\u00e9ponse est \u00e0 chercher du c\u00f4t\u00e9 des groupes de Coxeter&#8230;).<\/li>\n<\/ul>\n\n\t\t<style>\n\t\t\t#gallery-1 {\n\t\t\t\tmargin: auto;\n\t\t\t}\n\t\t\t#gallery-1 .gallery-item {\n\t\t\t\tfloat: left;\n\t\t\t\tmargin-top: 10px;\n\t\t\t\ttext-align: center;\n\t\t\t\twidth: 33%;\n\t\t\t}\n\t\t\t#gallery-1 img {\n\t\t\t\tborder: 2px solid #cfcfcf;\n\t\t\t}\n\t\t\t#gallery-1 .gallery-caption {\n\t\t\t\tmargin-left: 0;\n\t\t\t}\n\t\t\t\/* see gallery_shortcode() in wp-includes\/media.php *\/\n\t\t<\/style>\n\t\t<div id='gallery-1' class='gallery galleryid-1022 gallery-columns-3 gallery-size-thumbnail'><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2713\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2713-e1433344661104-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1024\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1024'>\n\t\t\t\tSomme de noeuds\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2719\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2719-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1030\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1030'>\n\t\t\t\tProduit de tresses\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2718\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2718-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1029\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1029'>\n\t\t\t\tProduit de tresses\n\t\t\t\t<\/dd><\/dl><br style=\"clear: both\" \/><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2717\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2717-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1028\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1028'>\n\t\t\t\tProduit de tresses\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2722\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2722-e1433344618741-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1033\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1033'>\n\t\t\t\tMouvements de Reidemeister\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2723\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2723-e1433344599993-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1034\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1034'>\n\t\t\t\tMouvements de Reidemeister\n\t\t\t\t<\/dd><\/dl><br style=\"clear: both\" \/><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2721\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2721-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1032\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1032'>\n\t\t\t\tMouvements de Reidemeister\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2724\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2724-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1035\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1035'>\n\t\t\t\tMouvements de Reidemeister\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2716\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2716-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1027\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1027'>\n\t\t\t\tInvariant de noeuds\n\t\t\t\t<\/dd><\/dl><br style=\"clear: both\" \/><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon portrait'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2728\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2728-e1433344578420-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1039\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1039'>\n\t\t\t\tInvariants de tresses\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2725\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2725-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1036\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1036'>\n\t\t\t\tNoeuds et ordinateur\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2726\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2726-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1037\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1037'>\n\t\t\t\tNoeuds et ordinateur\n\t\t\t\t<\/dd><\/dl><br style=\"clear: both\" \/><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2714\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2714-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1025\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1025'>\n\t\t\t\tNombre d&#8217;entrelacements\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2715\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2715-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1026\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1026'>\n\t\t\t\tNombre d&#8217;entrelacements\n\t\t\t\t<\/dd><\/dl><dl class='gallery-item'>\n\t\t\t<dt class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/2015\/06\/03\/lycee-saint-sernin-toulouse-tresses-et-entrelacs\/img_2727\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/files\/2015\/06\/IMG_2727-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" aria-describedby=\"gallery-1-1038\" \/><\/a>\n\t\t\t<\/dt>\n\t\t\t\t<dd class='wp-caption-text gallery-caption' id='gallery-1-1038'>\n\t\t\t\tProbl\u00e8me de miroirs\n\t\t\t\t<\/dd><\/dl><br style=\"clear: both\" \/>\n\t\t<\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Lundi, 9h : les \u00e9l\u00e8ves du lyc\u00e9e Saint-Sernin arrivent \u00e0 l&#8217;universit\u00e9 Paul Sabatier de Toulouse. Leur professeur de math\u00e9matiques les emm\u00e8ne \u00e0 l&#8217;institut de math\u00e9matiques o\u00f9\u00a0un enseignant-chercheur et trois doctorants sont l\u00e0 pour les accueillir. Le sujet sur lequel les \u00e9l\u00e8ves vont se casser la t\u00eate pendant trois jours est d\u00e9voil\u00e9 : ce sera les [&hellip;]<\/p>\n","protected":false},"author":131,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"gallery","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1022","post","type-post","status-publish","format-gallery","hentry","category-stages","post_format-post-format-gallery"],"_links":{"self":[{"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/posts\/1022","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/users\/131"}],"replies":[{"embeddable":true,"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/comments?post=1022"}],"version-history":[{"count":4,"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/posts\/1022\/revisions"}],"predecessor-version":[{"id":1044,"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/posts\/1022\/revisions\/1044"}],"wp:attachment":[{"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/media?parent=1022"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/categories?post=1022"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/perso.math.univ-toulouse.fr\/hippocampe\/wp-json\/wp\/v2\/tags?post=1022"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}