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Motivated by questions from (bio)-physical knot theory, the work introduces GridPyM, a Sage compatible Python module for generating and simplifying grid diagrams and for modelling local transformations between them, supporting efficient experiments beyond invariant computation.",{"@graph":69,"@context":121},[70,84,104],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/gridpym-a-python-module-to-handle-grid-diagrams-vol-14-2024-introduction/657600/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":98,"encodingFormat":97,"isAccessibleForFree":99,"interactionStatistic":100},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/gridpym-a-python-module-to-handle-grid-diagrams-vol-14-2024-introduction/657600.png","ImageObject",300,407,{"name":92,"@type":93},"SundayMiles","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-26",true,{"@type":101,"interactionType":102,"userInteractionCount":8},"InteractionCounter",{"@type":103},"ViewAction",{"@type":105,"mainEntity":106},"FAQPage",[107,113,117],{"name":108,"@type":109,"acceptedAnswer":110},"What is GridPyM and what problem does it address?","Question",{"text":111,"@type":112},"GridPyM is a Sage compatible Python module introduced to generate and simplify grid diagrams and to model local transformations between them, addressing practical needs in working with grid diagram representations.","Answer",{"name":114,"@type":109,"acceptedAnswer":115},"How do grid diagrams represent link diagrams?",{"text":116,"@type":112},"A grid diagram encodes an oriented link diagram by resolving each double point, and it provides a concise combinatorial description suitable for computational manipulation.",{"name":118,"@type":109,"acceptedAnswer":119},"Why is computing grid homology computationally difficult?",{"text":120,"@type":112},"A direct computation becomes quickly unfeasible because the associated chain complex has a number of generators that grows factorially with the grid size.","https://schema.org",{"og:url":83,"og:type":123,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":125,"canonical":83},"index,follow",{"doc_id":127,"site_id":62},657600,1793051987,{"code":4,"msg":5,"data":130},{"doc_id":127,"user_id":131,"nickname":92,"user_avatar":132,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":133,"file_id":134,"file_url":135,"file_type":136,"file_size":137,"view_count":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":52,"language":138,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":139,"faqs":140,"seo_title":141,"seo_description":67,"update_tm":142,"read_time":143},1099529839202,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Journal of Software for Algebra and Geometry  \nGridPyM: A Python module to handle grid diagrams  \nAGNESE BARBENSI AND DANIELE CELORIA  \nvol 14 2024  \nJSAG 14 (2024), 31–39 The Journal of Software for  \n[https://doi.org/10.2140/jsag.2024.14.31](https://doi.org/10.2140/jsag.2024.14.31) Algebra and Geometry  \nGridPyM: A Python module to handle grid diagrams  \nAGNESE BARBENSI AND DANIELE CELORIA  \nABSTRACT: Grid diagrams are a combinatorial version of classical link diagrams, widely used in theoretical, computational and applied knot theory. Motivated by questions from (bio)-physical knot theory, we introduce GridPyM, a Sage compatible Python module that handles grid diagrams. GridPyM focuses on generating and simplifying grids, and on modelling local transformations between them.  \n1. INTRODUCTION . Grid diagrams are a handy and computationally efficient model for oriented link diagrams. They admit a concise description, can be easily randomised and (unlike standard link diagrams) are able to encode certain local geometric features of links. They were first introduced in a slightly different form in [16], and later on in [9; 13] under the name of arc presentations. The latter paper also introduced a set of combinatorial moves, commonly known as Cromwell moves, connecting any two grids representing equivalent link diagrams.  \nGrid diagrams are ubiquitous in theoretical and applied knot theory. Indeed, they arise naturally in contact geometry, as they represent Legendrian diagrams of links in S3 , endowed with its standard tight contact structure. Further, a certain subset of Cromwell moves bijectively corresponds to Reidemeister moves preserving the Legendrian isotopy class of the link [14; 19] .  \nGrid diagrams can also be interpreted as representing links in a genus one Heegaard decomposition of the 3-sphere. Here, the grid is identified with the Heegaard torus, with horizontal and vertical lines acting as the cores and co-cores of the 1 and 2 handles. This is the ideal setup to define a combinatorial version of knot Floer homology [20; 24], known as grid homology [17]; see also [23] for a detailed reference, and [1] for a lens space version. Despite its straightforward definition, a direct computation of grid homology becomes quickly unfeasible, as the chain complex has a number of generators which grows factorially in the size of the grid. It is, however, possible to greatly simplify the grid homology chain complex [5], and there is a relatively efficient algorithm to compute it [2] implemented in the software GridLink [10] . Note that faster computation of knot Floer homology [22] can be achieved using more recent techniques developed in [21] .  \nThanks to their combinatorial definition, grid diagrams provide a convenient model to investigate on asymptotic properties of (bio)physical knots [12; 27] . As an example, they have been used to quantify the intensity of crossing change-mediated fluxes between different knot types, and to demonstrate the  \nMSC2020: primary 57K10; secondary 90-04 .  \nKeywords: knot theory, low-dimensional topology, computational topology.  \nGridPyM version 1.2  \n© 2024 The Authors, under license to MSP (Mathematical Sciences Publishers) .  \n32 Barbensi and Celoria :::: GridPyM: A Python module to handle grid diagrams  \nFigure 1. From left to right: an example of a grid diagram, and how to obtain an oriented link diagram from it by resolving each double point as a vertical overpass.  \ndependency of those fluxes on the local geometry of knotted configurations. In turn, this provided strong evidence that the simplification action of specific DNA enzymes is driven by a geometric selection of sites [4] . One further potential use of grid diagrams is to help with the search of band changes and the determination of Gordian distance between knot types [6; 7 ; 18 ; 26] .  \nThe most used program currently available to manipulate grid diagrams is GridLink. One of the main features of GridLink is a very efficient","cbCaiaunjnxgH31L","https://ap.wps.com/l/cbCaiaunjnxgH31L","pdf",1025199,"English","# Abstract\n# Introduction\n## Grid diagrams and their computational efficiency\n## Connections to contact geometry and Heegaard decompositions\n## Grid homology and simplification tools\n# Grid diagrams definition and representation","[{\"question\":\"What is GridPyM and what problem does it address?\",\"answer\":\"GridPyM is a Sage compatible Python module introduced to generate and simplify grid diagrams and to model local transformations between them, addressing practical needs in working with grid diagram representations.\"},{\"question\":\"How do grid diagrams represent link diagrams?\",\"answer\":\"A grid diagram encodes an oriented link diagram by resolving each double point, and it provides a concise combinatorial description suitable for computational manipulation.\"},{\"question\":\"Why is computing grid homology computationally difficult?\",\"answer\":\"A direct computation becomes quickly unfeasible because the associated chain complex has a number of generators that grows factorially with the grid size.\"}]","GridPyM: A Python module to handle grid diagrams - vol 14 2024 - introduction | PDF",1792984587,25]