{"id":81317,"date":"2024-11-20T12:47:39","date_gmt":"2024-11-20T11:47:39","guid":{"rendered":"https:\/\/wfa.uwr.edu.pl\/?p=81317"},"modified":"2024-11-20T12:47:44","modified_gmt":"2024-11-20T11:47:44","slug":"study-of-the-knot-quiver-correspondence","status":"publish","type":"post","link":"https:\/\/wfa.uwr.edu.pl\/en\/2024\/11\/20\/study-of-the-knot-quiver-correspondence\/","title":{"rendered":"Study of the knot-quiver correspondence"},"content":{"rendered":"\n<div class=\"wp-block-ugb-container ugb-container blok_nowa_podstrona ugb-8dda971 ugb-container--v2 ugb-container--design-plain ugb-main-block\"><div class=\"ugb-inner-block\"><div class=\"ugb-block-content\"><div class=\"ugb-container__wrapper ugb-8dda971-wrapper\"><div class=\"ugb-container__side\"><div class=\"ugb-container__content-wrapper ugb-8dda971-content-wrapper\">\n<div class=\"wp-block-columns bs-news-page is-layout-flex wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column bs-kontakt-photo is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"299\" height=\"168\" src=\"https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel.jpg\" alt=\"w\u0119ze\u0142\" class=\"wp-image-53843\" srcset=\"https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel.jpg 299w, https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel-24x13.jpg 24w, https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel-36x20.jpg 36w, https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel-48x27.jpg 48w\" sizes=\"auto, (max-width: 299px) 100vw, 299px\" \/><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column bs-contact-right-column is-layout-flow wp-block-column-is-layout-flow\">\n<div class=\"wp-block-bonasoft-modular-breadcrumbs-block\"><div class=\"bs_add_breadcrumb_trail\"><\/div><\/div>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-black-color has-css-opacity has-black-background-color has-background is-style-wide bs-border-kontakt\"\/>\n\n\n<h1 style=\"font-style:normal;font-weight:700;\" class=\"bs-naglowek-podstrony wp-block-post-title\">Study of the knot-quiver correspondence<\/h1>\n\n\n<p>In the paper titled <em>Permutohedra for knots and quivers <\/em>published by Jakub Jankowski, Piotr Kucharski, <em>Helder Larragu\u00edvel<\/em>, Dmitry Noshchenko, and Piotr Su\u0142kowski in the journal Phys. Rev. D 104, 086017 (2021) the authors study the knot-quiver correspondence.<\/p>\n\n\n<div class=\"bs__links\">\r\n    <div class=\"bs__links__section\">\r\n        <ul class=\"bs__links__list\">\r\n                            <li class=\"bs__links__list__item\">\r\n                    <a href=\"https:\/\/journals.aps.org\/prd\/abstract\/10.1103\/PhysRevD.104.086017?ft=1#fulltext\" class=\"bs__links__list__item__link\">\r\n                        \u201ePermutohedra for knots and quivers\u201d \r\n                        <div class=\"bs__links__list__item__link__before__icon\">\r\n                            <div class=\"bs__links__list__item__link__icon\">\r\n                                <svg width=\"40px\" height=\"40px\" viewBox=\"0 0 22 16\" version=\"1.1\">\r\n                                <g stroke=\"none\" strokeWidth=\"1\" fill=\"black\" fillRule=\"evenodd\">\r\n                                    <g>\r\n                                    <polygon points=\"13.4875 0 12 1.6215 18.7875 8 12 14.3795 13.4875 16 21.9995 8\" \/>\r\n                                    <\/g>\r\n                                <\/g>\r\n                                <\/svg>\r\n                            <\/div>\r\n                        <\/div>\r\n                    <\/a>\r\n                <li>\r\n                            <li class=\"bs__links__list__item\">\r\n                    <a href=\"https:\/\/wfa.uwr.edu.pl\/lista-pracownikow\/?omegaId=UWRa2dc468d8d444704b452c82f5f763f7c\" class=\"bs__links__list__item__link\">\r\n                        Jakub Jankowski \r\n                        <div class=\"bs__links__list__item__link__before__icon\">\r\n                            <div class=\"bs__links__list__item__link__icon\">\r\n                                <svg width=\"40px\" height=\"40px\" viewBox=\"0 0 22 16\" version=\"1.1\">\r\n                                <g stroke=\"none\" strokeWidth=\"1\" fill=\"black\" fillRule=\"evenodd\">\r\n                                    <g>\r\n                                    <polygon points=\"13.4875 0 12 1.6215 18.7875 8 12 14.3795 13.4875 16 21.9995 8\" \/>\r\n                                    <\/g>\r\n                                <\/g>\r\n                                <\/svg>\r\n                            <\/div>\r\n                        <\/div>\r\n                    <\/a>\r\n                <li>\r\n                    <\/ul>\r\n    <\/div>\r\n<\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-ugb-container ugb-container bs-news-page-under ugb-f8694ce ugb-container--v2 ugb-container--design-plain ugb-main-block\"><div class=\"ugb-inner-block\"><div class=\"ugb-block-content\"><div class=\"ugb-container__wrapper ugb-f8694ce-wrapper\"><div class=\"ugb-container__side\"><div class=\"ugb-container__content-wrapper ugb-f8694ce-content-wrapper\">\n<figure class=\"wp-block-image aligncenter size-full\"><img decoding=\"async\" src=\"https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2022\/08\/Trefoil-knot-and-the-corresponding-quiver.png\" alt=\"Rys. 1 W\u0119ze\u0142 tr\u00f3j-listnik (z lewej), posiadaj\u0105cy trzy przeci\u0119cia. Z prawej odpowiadaj\u0105cy mu ko\u0142czan z\u0142o\u017cony z trzech wierzcho\u0142k\u00f3w oraz zestawu kraw\u0119dzi i p\u0119tli.\" class=\"wp-image-32776\"\/><figcaption class=\"wp-element-caption\">Fig. 1 Trefoil knot (left) with three intersections. On the right a corresponding quiver, consisting of three peaks and a set of edges and loops.<\/figcaption><\/figure>\n\n\n\n<p>Knots are closed loops placed in three-dimensional space, characterised by the number of intersections. For instance, the knot in Fig. 1 has three intersections and is the simplest non-trivial knot, and the only one with three intersections. Due to its shape it is called a trefoil.The fundamental and unresolved issue of knot theory is their classification and distinction that mathematicians carry out by assigning to knots various objects called topological invariants. Over decades a multitude of various kinds of invariants of larger or lesser use appeared, but the ideal one that would help to distinguish any two knots remains elusive.<\/p>\n\n\n<div class=\"bs__links\">\r\n    <div class=\"bs__links__section\">\r\n        <ul class=\"bs__links__list\">\r\n                            <li class=\"bs__links__list__item\">\r\n                    <a href=\" http:\/\/katlas.org\/wiki\/Main_Page\" class=\"bs__links__list__item__link\">\r\n                        The knowledge obtained in this way is available in the catalogue of knots on the website  \r\n                        <div class=\"bs__links__list__item__link__before__icon\">\r\n                            <div class=\"bs__links__list__item__link__icon\">\r\n                                <svg width=\"40px\" height=\"40px\" viewBox=\"0 0 22 16\" version=\"1.1\">\r\n                                <g stroke=\"none\" strokeWidth=\"1\" fill=\"black\" fillRule=\"evenodd\">\r\n                                    <g>\r\n                                    <polygon points=\"13.4875 0 12 1.6215 18.7875 8 12 14.3795 13.4875 16 21.9995 8\" \/>\r\n                                    <\/g>\r\n                                <\/g>\r\n                                <\/svg>\r\n                            <\/div>\r\n                        <\/div>\r\n                    <\/a>\r\n                <li>\r\n                    <\/ul>\r\n    <\/div>\r\n<\/div>\n\n\n<p>A huge surprise was the discovery of correspondence between certain physical values and knot invariants in the late 1980s by Edward Witten. Inspirations from physics have been an important branch in development of knot theory ever since, generating significant progress in mathematics. Part of this progress was characterising a whole array of invariants with the so-called quivers. Quivers in mathematics are something else than what Robin Hood used; they are graphs composed of a finite number of peaks connected with a defined number of arrows. It turns out that such objects are incredibly useful in defining knot theory invariants and that each knot seems to have a corresponding quiver. For instance, the simplest knot in Fig. 1 is assigned to a quiver composed of three peaks and a set of lines (the convergence of the number of the intersections of the knot and the peaks is coincidental).<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img decoding=\"async\" src=\"https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2022\/08\/3DGraph91.png\" alt=\"Rys. 2 R\u00f3wnowa\u017cne ko\u0142czany dla w\u0119z\u0142a o dziewi\u0119ciu przeci\u0119ciach. Ka\u017cda niebieska kropka oznacza jeden ko\u0142czan, a kolorowe po\u0142\u0105czenia operacje symetrii mi\u0119dzy ko\u0142czanami.\" class=\"wp-image-32782\" style=\"width:495px;height:175px\"\/><figcaption class=\"wp-element-caption\">Fig. 2\u00a0 Equivalent quivers for a knot with nine intersections. Each blue dot denotes one quiver while the colourful lines mark operations of symmetries between the quivers.<\/figcaption><\/figure>\n\n\n\n<p>In the described paper the task was to study whether the knot-quiver correspondence is unambiguous, i.e. whether each knot has exactly one assigned quiver, and if not then is there an equivalent quiver structure? To our surprise it turned out that not only the correspondence of knots and quivers is not unambiguous, but also that a very large number of equivalent quivers may correspond to a single knot! For instance, for knots with six intersections this number reaches 100 000! Moreover, it turns out that equivalent quivers display a beautiful geometrical structure that is a multi-dimensional generalisation of a regular polyhedron, an example of which is in Fig. 2, presenting equivalent quivers for a knot with nine intersections. The meaning of those structures and their consequences for the physical as well as mathematical aspects of the issue are the topic of current studies.\u00a0<\/p>\n\n\n\n<p>Added by: Joanna Molenda-\u017bakowicz<\/p>\n\n\n\n<p>Dean&#8217;s representative for promotion and media relations<\/p>\n<\/div><\/div><\/div><\/div><\/div><\/div>\n<\/div><\/div><\/div><\/div><\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>In the paper titled Permutohedra for knots and quivers published by Jakub Jankowski, Piotr Kucharski, Helder Larragu\u00edvel, Dmitry Noshchenko, and Piotr Su\u0142kowski in the journal Phys. Rev. D 104, 086017 (2021) the authors study the knot-quiver correspondence. Knots are closed loops placed in three-dimensional space, characterised by the number of intersections. For instance, the knot [&hellip;]<\/p>\n","protected":false},"author":7182,"featured_media":53843,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[389],"tags":[392],"class_list":["post-81317","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-publikacje-en","tag-publikacje-en"],"featured_image_urls_v2":{"full":["https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel.jpg",299,168,false],"thumbnail":["https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel-150x150.jpg",150,150,true],"medium":["https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel.jpg",299,168,false],"medium_large":["https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel.jpg",299,168,false],"large":["https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel.jpg",299,168,false],"1536x1536":["https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel.jpg",299,168,false],"2048x2048":["https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel.jpg",299,168,false],"menu-24x24":["https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel-24x13.jpg",24,13,true],"menu-36x36":["https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel-36x20.jpg",36,20,true],"menu-48x48":["https:\/\/wfa.uwr.edu.pl\/wp-content\/uploads\/sites\/216\/2023\/07\/wezel-48x27.jpg",48,27,true]},"post_excerpt_stackable_v2":"<p>Study of the knot-quiver correspondence In the paper titled Permutohedra for knots and quivers published by Jakub Jankowski, Piotr Kucharski, Helder Larragu\u00edvel, Dmitry Noshchenko, and Piotr Su\u0142kowski in the journal Phys. Rev. D 104, 086017 (2021) the authors study the knot-quiver correspondence. \u201ePermutohedra for knots and quivers\u201d Jakub Jankowski Fig. 1 Trefoil knot (left) with three intersections. On the right a corresponding quiver, consisting of three peaks and a set of edges and loops. Knots are closed loops placed in three-dimensional space, characterised by the number of intersections. For instance, the knot in Fig. 1 has three intersections and is&hellip;<\/p>\n","category_list_v2":"<a href=\"https:\/\/wfa.uwr.edu.pl\/en\/category\/publikacje-en\/\" rel=\"category tag\">Publications<\/a>","author_info_v2":{"name":"kswistak","url":"https:\/\/wfa.uwr.edu.pl\/en\/author\/kswistak\/"},"comments_num_v2":"0 comments","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Study of the knot-quiver correspondence - Faculty of Physics and Astronomy<\/title>\n<meta name=\"description\" content=\"Wydzia\u0142 Fizyki i Astronomii\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/wfa.uwr.edu.pl\/en\/2024\/11\/20\/study-of-the-knot-quiver-correspondence\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Study of the knot-quiver correspondence - 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