[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125395-en":3,"doc-seo-125395-105":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},125395,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Machine Learning of Knot Topology in Non-Hermitian Band Braids","Deep relationships between braids, knots, and topological physics enable classification of topological states across physical platforms, but distinguishing braid groups and knot topology in non-Hermitian settings remains difficult. An unsupervised machine-learning method using su(n) Lie-algebra representation bases on n-fold extended non-Hermitian bands classifies braid group and knot topology without prior mathematical knowledge or predefined topological invariants. It identifies unlink, unknot, Hopf link, Solomon ring, trefoil, and more, and—by including eigenstate information—separates parity-time phases, opposite chiralities, and previously overlooked topological phases.","arXiv :2401 . 10908v1 [ cond-mat .mes-hall ] 8 Jan 2024  \nMachine Learning of Knot Topology in Non-Hermitian Band Braids  \nJiangzhi Chen, 1, ∗ Zi Wang, 1, ∗ Yu-Tao Tan, 1 Ce Wang, 1,† and Jie Ren 1, 2,‡ 1 Center for Phononics and Thermal Energy Science, China-EU Joint Lab on Nanophononics,  \nShanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 2 Shanghai Research Institute for Intelligent Autonomous Systems, Tongji University, Shanghai 200092, P. R. China  \nThe deep connection among braids, knots and topological physics has provided valuable insights into studying topological states in various physical systems. However, identifying distinct braid groups and knot topology embedded in non-Hermitian systems is challenging and requires significant efforts. Here, we demonstrate that an unsupervised learning with the representation basis of su(n) Lie algebra on n-fold extended non-Hermitian bands can fully classify braid group and knot topology therein, without requiring any prior mathematical knowledge or any pre-defined topological invariants. We demonstrate that the approach successfully identifies different topological elements, such as unlink, unknot, Hopf link, Solomon ring, trefoil, and so on, by employing generalized Gell-Mann matrices in non-Hermitian models with n=2 and n=3 energy bands. Moreover, since eigenstate information of non-Hermitian bands is incorporated in addition to eigenvalues, the approach distinguishes the different parity-time symmetry and breaking phases, recognizes the opposite chirality of braids and knots, and identifies out distinct topological phases that were overlooked before. Our study shows significant potential of machine learning in classification of knots, braid groups, and non-Hermitian topological phases.  \nIntroduction-Knot theory holds special significance in science [1], which provides a crucial mathematical language to understand the topological properties and interactions of various physical systems [2–8] . For example, the knot invariants derived from solutions of the Yang-Baxter equation are used to characterize the entanglement and topological properties of quantum states [9, 10] . Furthermore, knot theory contributes to the exploration of topological matters, such as topological insulators [11] and topological superconductors [12], which exhibit exotic phases and protected properties. Recently, it has been discovered that the distinct topological phases ofthe non-Hermitian band structure in reciprocal space can be classified with braid groups, which are closely related to knots [13–21] . Concrete models that can exhibit different braid patterns have been experimentally implemented in platforms such as optical systems [17–19], quantum circuits [20] and ideal acoustic metamaterials [4] . Band braiding structure is also recently shown to induce topologically protected quantized response [22] . Closed braids represent knots, which are well-established in mathematics.  \nYet, there remain great challenges for knot theory, both mathematically and physically in the specific context of nonHermitian band braids. One of these challenges is the classification and identification of knot types. To overcome the challenge mathematically, the Alexander polynomial was first proposed back in 1923, and subsequently, various other polynomial forms were introduced [23, 24] . However, the algebraicbased methods used to calculate knot invariants are computationally complex. Furthermore, relying solely on one or a few invariants poses challenges in achieving a comprehensive classification of knots. With the rapid development of artificial intelligence, supervised neural networks [25, 26] are applied to those knot topology. However, this approach relies on tremendous training samples with prior knowledge of known knot types, which necessitates substantial computational resources, and explori","cbCaiscSCB2UlFva","https://ap.wps.com/l/cbCaiscSCB2UlFva","pdf",6984993,1,7,"English","en",105,"# Introduction\n## Knot theory and topological physics background\n## Challenges of classifying non-Hermitian knot types\n## Existing approaches and their limitations\n## Non-Hermitian band braids: eigenvalues and eigenvectors","[{\"question\":\"What problem does the study address in non-Hermitian band braids?\",\"answer\":\"It addresses the difficulty of identifying distinct braid groups and knot topology embedded in non-Hermitian systems, which requires substantial effort in existing approaches.\"},{\"question\":\"How does the proposed method classify knot topology and braid groups?\",\"answer\":\"It uses an unsupervised learning scheme based on the representation basis of su(n) Lie algebra on n-fold extended non-Hermitian bands, without requiring prior mathematical knowledge or predefined topological invariants.\"},{\"question\":\"What additional phases can the approach distinguish beyond knot topology?\",\"answer\":\"By incorporating eigenstate information in addition to eigenvalues, it distinguishes parity-time symmetry and breaking phases and recognizes opposite chirality of braids and knots, identifying topological phases that were previously overlooked.\"}]","Machine Learning of Knot Topology in Non-Hermitian Band Braids | 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problem does the study address in non-Hermitian band braids?","Question",{"text":75,"@type":76},"It addresses the difficulty of identifying distinct braid groups and knot topology embedded in non-Hermitian systems, which requires substantial effort in existing approaches.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method classify knot topology and braid groups?",{"text":80,"@type":76},"It uses an unsupervised learning scheme based on the representation basis of su(n) Lie algebra on n-fold extended non-Hermitian bands, without requiring prior mathematical knowledge or predefined topological invariants.",{"name":82,"@type":73,"acceptedAnswer":83},"What additional phases can the approach distinguish beyond knot topology?",{"text":84,"@type":76},"By incorporating eigenstate information in addition to eigenvalues, it distinguishes parity-time symmetry and breaking phases and recognizes opposite chirality of braids and knots, identifying topological phases 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