{"id":14128,"date":"2025-09-09T17:31:46","date_gmt":"2025-09-09T12:01:46","guid":{"rendered":"\/maths\/?p=14128"},"modified":"2026-04-09T16:57:50","modified_gmt":"2026-04-09T11:27:50","slug":"dr-m-prabhakar-2","status":"publish","type":"post","link":"\/maths\/?p=14128","title":{"rendered":"Dr. M. Prabhakar"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"14128\" class=\"elementor elementor-14128\" data-elementor-post-type=\"post\">\n\t\t\t\t<div class=\"elementor-element elementor-element-e322a2b e-con-full e-flex e-con e-parent\" data-id=\"e322a2b\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t<div class=\"elementor-element elementor-element-133d579 e-con-full e-flex e-con e-child\" data-id=\"133d579\" data-element_type=\"container\">\n\t\t<div class=\"elementor-element elementor-element-eb40695 e-con-full e-flex e-con e-child\" data-id=\"eb40695\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t<div class=\"elementor-element elementor-element-e0630dc e-con-full e-flex e-con e-child\" data-id=\"e0630dc\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5d1d5ed elementor-widget elementor-widget-image\" data-id=\"5d1d5ed\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"658\" height=\"634\" src=\"\/maths\/wp-content\/uploads\/2025\/09\/Me.jpg\" class=\"attachment-full size-full wp-image-14221\" alt=\"\" srcset=\"\/maths\/wp-content\/uploads\/2025\/09\/Me.jpg 658w, \/maths\/wp-content\/uploads\/2025\/09\/Me-300x289.jpg 300w\" sizes=\"(max-width: 658px) 100vw, 658px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-f7ac483 e-flex e-con-boxed e-con e-child\" data-id=\"f7ac483\" data-element_type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-609cbc9 elementor-widget elementor-widget-heading\" data-id=\"609cbc9\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h4 class=\"elementor-heading-title elementor-size-default\">Dr. M. Prabhkar<\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7db05b5 elementor-widget elementor-widget-heading\" data-id=\"7db05b5\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h5 class=\"elementor-heading-title elementor-size-default\">Associate Professor<\/h5>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-23da870 elementor-icon-list--layout-traditional elementor-list-item-link-full_width elementor-widget elementor-widget-icon-list\" data-id=\"23da870\" data-element_type=\"widget\" data-widget_type=\"icon-list.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul class=\"elementor-icon-list-items\">\n\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Department of Mathematics<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Indian Institute of Technology Ropar<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Rupnagar, Punjab - 140001, India<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t<\/ul>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1c383cd elementor-icon-list--layout-traditional elementor-list-item-link-full_width elementor-widget elementor-widget-icon-list\" data-id=\"1c383cd\" data-element_type=\"widget\" data-widget_type=\"icon-list.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul class=\"elementor-icon-list-items\">\n\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-address-card\" viewBox=\"0 0 576 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M528 32H48C21.5 32 0 53.5 0 80v352c0 26.5 21.5 48 48 48h480c26.5 0 48-21.5 48-48V80c0-26.5-21.5-48-48-48zm-352 96c35.3 0 64 28.7 64 64s-28.7 64-64 64-64-28.7-64-64 28.7-64 64-64zm112 236.8c0 10.6-10 19.2-22.4 19.2H86.4C74 384 64 375.4 64 364.8v-19.2c0-31.8 30.1-57.6 67.2-57.6h5c12.3 5.1 25.7 8 39.8 8s27.6-2.9 39.8-8h5c37.1 0 67.2 25.8 67.2 57.6v19.2zM512 312c0 4.4-3.6 8-8 8H360c-4.4 0-8-3.6-8-8v-16c0-4.4 3.6-8 8-8h144c4.4 0 8 3.6 8 8v16zm0-64c0 4.4-3.6 8-8 8H360c-4.4 0-8-3.6-8-8v-16c0-4.4 3.6-8 8-8h144c4.4 0 8 3.6 8 8v16zm0-64c0 4.4-3.6 8-8 8H360c-4.4 0-8-3.6-8-8v-16c0-4.4 3.6-8 8-8h144c4.4 0 8 3.6 8 8v16z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Office:  C-M28, Mezzanine Floor, SAB building<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-phone-square-alt\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M400 32H48A48 48 0 0 0 0 80v352a48 48 0 0 0 48 48h352a48 48 0 0 0 48-48V80a48 48 0 0 0-48-48zm-16.39 307.37l-15 65A15 15 0 0 1 354 416C194 416 64 286.29 64 126a15.7 15.7 0 0 1 11.63-14.61l65-15A18.23 18.23 0 0 1 144 96a16.27 16.27 0 0 1 13.79 9.09l30 70A17.9 17.9 0 0 1 189 181a17 17 0 0 1-5.5 11.61l-37.89 31a231.91 231.91 0 0 0 110.78 110.78l31-37.89A17 17 0 0 1 299 291a17.85 17.85 0 0 1 5.91 1.21l70 30A16.25 16.25 0 0 1 384 336a17.41 17.41 0 0 1-.39 3.37z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Phone: 0188123-2317 (O)<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<a href=\"mailto:parthasharathi@iitrpr.ac.in\">\n\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-envelope\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Email : prabhakar@iitrpr.ac.in<\/span>\n\t\t\t\t\t\t\t\t\t\t\t<\/a>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t<\/ul>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-db20d41 e-flex e-con-boxed e-con e-child\" data-id=\"db20d41\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-6716fad e-con-full e-flex e-con e-parent\" data-id=\"6716fad\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-942c71d main-tabs e-n-tabs-mobile elementor-widget elementor-widget-n-tabs\" data-id=\"942c71d\" data-element_type=\"widget\" data-widget_type=\"nested-tabs.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"e-n-tabs\" data-widget-number=\"155371293\" aria-label=\"Tabs. Open items with Enter or Space, close with Escape and navigate using the Arrow keys.\">\n\t\t\t<div class=\"e-n-tabs-heading\" role=\"tablist\">\n\t\t\t\t\t<button id=\"e-n-tab-title-1553712931\" class=\"e-n-tab-title\" aria-selected=\"true\" data-tab-index=\"1\" role=\"tab\" tabindex=\"0\" aria-controls=\"e-n-tab-content-1553712931\" style=\"--n-tabs-title-order: 1;\">\n\t\t\t\t\t\t<span class=\"e-n-tab-title-text\">\n\t\t\t\tAbout\t\t\t<\/span>\n\t\t<\/button>\n\t\t\t\t<button id=\"e-n-tab-title-1553712932\" class=\"e-n-tab-title\" aria-selected=\"false\" data-tab-index=\"2\" role=\"tab\" tabindex=\"-1\" aria-controls=\"e-n-tab-content-1553712932\" style=\"--n-tabs-title-order: 2;\">\n\t\t\t\t\t\t<span class=\"e-n-tab-title-text\">\n\t\t\t\tEducation\t\t\t<\/span>\n\t\t<\/button>\n\t\t\t\t<button id=\"e-n-tab-title-1553712933\" class=\"e-n-tab-title\" aria-selected=\"false\" data-tab-index=\"3\" role=\"tab\" tabindex=\"-1\" aria-controls=\"e-n-tab-content-1553712933\" style=\"--n-tabs-title-order: 3;\">\n\t\t\t\t\t\t<span class=\"e-n-tab-title-text\">\n\t\t\t\tWork Experience\t\t\t<\/span>\n\t\t<\/button>\n\t\t\t\t<button id=\"e-n-tab-title-1553712934\" class=\"e-n-tab-title\" aria-selected=\"false\" data-tab-index=\"4\" role=\"tab\" tabindex=\"-1\" aria-controls=\"e-n-tab-content-1553712934\" style=\"--n-tabs-title-order: 4;\">\n\t\t\t\t\t\t<span class=\"e-n-tab-title-text\">\n\t\t\t\tPublications\t\t\t<\/span>\n\t\t<\/button>\n\t\t\t\t<button id=\"e-n-tab-title-1553712935\" class=\"e-n-tab-title\" aria-selected=\"false\" data-tab-index=\"5\" role=\"tab\" tabindex=\"-1\" aria-controls=\"e-n-tab-content-1553712935\" style=\"--n-tabs-title-order: 5;\">\n\t\t\t\t\t\t<span class=\"e-n-tab-title-text\">\n\t\t\t\tOther Information\t\t\t<\/span>\n\t\t<\/button>\n\t\t\t\t\t<\/div>\n\t\t\t<div class=\"e-n-tabs-content\">\n\t\t\t\t<div id=\"e-n-tab-content-1553712931\" role=\"tabpanel\" aria-labelledby=\"e-n-tab-title-1553712931\" data-tab-index=\"1\" style=\"--n-tabs-title-order: 1;\" class=\"e-active elementor-element elementor-element-7c001a3 e-con-full e-flex e-con e-child\" data-id=\"7c001a3\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-7a0ac13 elementor-widget elementor-widget-text-editor\" data-id=\"7a0ac13\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\tDr Prabhakar is an Associate Professor in the Department of Mathematics at IIT Ropar. After obtaining his doctoral degree from IIT Delhi, he carried out post-doctoral studies at Harish Chandra Research Institute, Allahabad. He was also a visiting research fellow at Osaka City University (Japan). Dr. Prabhakar was a faculty member in the Department of Mathematics at IIT Guwahati from March 2006 to July 2009.\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d249f5a elementor-widget elementor-widget-text-editor\" data-id=\"d249f5a\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"field-label\"><div class=\"field-label\"><strong>Areas of Research:\u00a0<\/strong><\/div><div class=\"field-items\"><ol><li class=\"field-item even\">Knot Theory<\/li><li class=\"field-item odd\">Spatial Graphs<\/li><\/ol><\/div><\/div>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div id=\"e-n-tab-content-1553712932\" role=\"tabpanel\" aria-labelledby=\"e-n-tab-title-1553712932\" data-tab-index=\"2\" style=\"--n-tabs-title-order: 2;\" class=\" elementor-element elementor-element-4116995 e-con-full e-flex e-con e-child\" data-id=\"4116995\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-f764086 elementor-icon-list--layout-traditional elementor-list-item-link-full_width elementor-widget elementor-widget-icon-list\" data-id=\"f764086\" data-element_type=\"widget\" data-widget_type=\"icon-list.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul class=\"elementor-icon-list-items\">\n\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Ph.D., Indian Institute of Technology Delhi, India, 2005<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">M.Sc., Andhra University, India, 1998<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">B.Sc., Andhra University, India, 1996<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t<\/ul>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div id=\"e-n-tab-content-1553712933\" role=\"tabpanel\" aria-labelledby=\"e-n-tab-title-1553712933\" data-tab-index=\"3\" style=\"--n-tabs-title-order: 3;\" class=\" elementor-element elementor-element-be9689c e-con-full e-flex e-con e-child\" data-id=\"be9689c\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-3b9b882 elementor-icon-list--layout-traditional elementor-list-item-link-full_width elementor-widget elementor-widget-icon-list\" data-id=\"3b9b882\" data-element_type=\"widget\" data-widget_type=\"icon-list.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul class=\"elementor-icon-list-items\">\n\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Associate Professor, Indian Institute of Technology Ropar, India, December 2015 \u2013 at present<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Assistant Professor, Indian Institute of Technology Ropar, India, 2009 \u2013 December 2015<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Senior Lecturer, Indian Institute of Technology Guwahati, India, 2006-09<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Visiting Researcher, Osaka City University, Osaka, Japan, 2006<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Post-Doctoral Fellow, HRI, Allahabad, India, 2005-06<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-angle-right\" viewBox=\"0 0 256 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">Teaching Assistant, Indian Institute of Technology Delhi, India, 2000-05<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t<\/ul>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div id=\"e-n-tab-content-1553712934\" role=\"tabpanel\" aria-labelledby=\"e-n-tab-title-1553712934\" data-tab-index=\"4\" style=\"--n-tabs-title-order: 4;\" class=\" elementor-element elementor-element-13f8258 e-con-full e-flex e-con e-child\" data-id=\"13f8258\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t<div class=\"elementor-element elementor-element-970688f e-flex e-con-boxed e-con e-child\" data-id=\"970688f\" data-element_type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-a54f3a2 elementor-icon-list--layout-traditional elementor-list-item-link-full_width elementor-widget elementor-widget-icon-list\" data-id=\"a54f3a2\" data-element_type=\"widget\" data-widget_type=\"icon-list.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul class=\"elementor-icon-list-items\">\n\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">1.\tKomal Negi, Ayaka Shimizu, Yoshiro Yaguchi, and Madeti Prabhakar, Orbits by the up\u2013down action of braid diagrams, Journal of Knot Theory and Its Ramifications, Vol. 34, No. 08, 2550031, 2025<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">2.\tK. Negi, A. Shimizu, M. Prabhakar, Warping Labeling for twisted knots and twisted virtual braids, Tokyo Journal of Mathematics, Vol. 48, No. 2, 2025<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">3.\tV. Bardakov, T. Kozlovskaya, K. Negi, and M. Prabhakar, Twisted virtual braid group, Journal of knot theory and its ramifications, (https:\/\/doi.org\/10.1142\/S0218216525500282), 2025<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">4.\tK. Negi and M. Prabhakar, Generalization of Arc Shift for twisted knots, Journal of knot theory and its ramifications, Vol. 33, No. 02, 2450004, 2024<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">5.\tK. Negi and M. Prabhakar, Singular twisted links and Singular twisted virtual braids, International Journal of Mathematics, (https:\/\/doi.org\/10.1142\/S0129167X25500211), 2025.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">6.\tKirandeep Kaur, and Prabhakar Madeti, Virtual Knots and Links with Unknotting index (n,m), Bulletin of the Korean Mathematical Society, (https:\/\/doi.org\/10.4134\/BKMS.b240494) 2025.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">7.\tK. Negi, M. Prabhakar, and S. Kamada, \u201cTwisted virtual braids and twisted links,\u201d Osaka Journal of Mathematics, vol. 61-4, 2024.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">8.\t S. Joshi, K. Negi, and M. Prabhakar, \u201cSome evaluations of the Jones polynomial for certain families of weaving knots,\u201d Topology Appl., vol. 329, Paper No. 108466, 11, 2023.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">K. Negi and M. Prabhakar, \u201cNumerical invariants for two-component virtual spatial graphs,\u201d Journal of Topology and Analysis, vol. Online Ready, Paper No. 1\u201313, (doi: 10.1142\/S1793525323500310), 2023.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">10.\tA. Gill, M. Ivanov, M. Prabhakar, and A. Vesnin, \u201cRecurrent generalization of f-polynomials for virtual knots and links,\u201d Symmetry, vol. 14, no. 1, pp. 1\u201315, (doi: 10.3390\/sym14010015), 2022.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">11.\tA. Gill and P. Madeti, \u201cVariations in writhes of virtual knots under a local move,\u201d Bull. Korean Math. Soc., vol. 59, no. 2, pp. 303\u2013318, 2022, issn: 1015-8634,2234-3016. doi: 10.4134\/BKMS.b200927.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">12.\tS. Joshi and M. Prabhakar, \u201cDeterminants of twisted generalized hybrid weaving knots,\u201d J. Knot Theory Ramifications, vol. 31, no. 14, Paper No. 2250104, 10, 2022, issn: 0218-2165,1793-6527.  doi: 10.1142\/S0218216522501048. <\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">13.\tA. Gill, K. Kaur, M. Prabhakar, and A. Vesnin, \u201cAn unknotting invariant for welded knots,\u201d Proc. Indian Acad. Sci. Math. Sci., vol. 131, no. 2, Paper No. 47, 17, 2021, issn: 0253-4142,0973-7685. doi: 10.1007\/s12044-021-00640-9.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">14.\tS. Joshi and M. Prabhakar, \u201cThe Gordian complex of theta-curves,\u201d J. Knot Theory Ramifications, vol. 30, no. 8, Paper No. 2150050, 11, 2021, issn: 0218-2165,1793-6527. doi: 10.1142\/S0218216521500504.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">15.\tA. Gill, M. Prabhakar, and A. Vesnin, \u201cGordian complexes of knots and virtual knots given by region crossing changes and arc shift moves,\u201d J. Knot Theory Ramifications, vol. 29, no. 10, pp. 2042008, 24, 2020, issn: 0218-2165,1793-6527. doi: 10.1142\/S0218216520420080.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">16.\tA. Gill, K. Kaur, and P. Madeti, \u201cArc shift number and region arc shift number for virtual knots,\u201d J. Korean Math. Soc., vol. 56, no. 4, pp. 1063\u20131081, 2019, issn: 0304-9914,2234-3008. doi: 10.4134\/JKMS.j180615.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">17.\tK. Kaur, M. Prabhakar, and A. Vesnin, \u201cAn unknotting index for virtual links,\u201d Topology Appl., vol. 264, pp. 352\u2013368, 2019, issn: 0166-8641,1879-3207. doi: 10.1016\/j.topol.2019.06.030.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">18.\tK. Kaur, S. Kamada, A. Kawauchi, and P. Madeti, \u201cAn unknotting index for virtual knots,\u201d Tokyo J. Math., vol. 42, no. 2, pp. 357\u2013370, 2019, issn: 0387-3870. doi: 10.3836\/tjm\/1502179283.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">19.\tK. Kaur, M. Prabhakar, and A. Vesnin, \u201cTwo-variable polynomial invariants of virtual knots arising from flat virtual knot invariants,\u201d J. Knot Theory Ramifications, vol. 27, no. 13, pp. 1842015, 22, 2018, issn: 0218-2165,1793-6527. doi: 10.1142\/S0218216518420154.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">20.\tK. Kaur, S. Kamada, A. Kawauchi, and M. Prabhakar, \u201cGauss diagrams, unknotting numbers and trivializing numbers of spatial graphs,\u201d Topology Appl., vol. 230, pp. 586\u2013598, 2017, issn: 0166-8641,1879-3207. doi: 10.1016\/j.topol.2017.08.037.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">21.\tV. Siwach and M. Prabhakar, \u201cOn minimal unknotting crossing data for closed toric braids,\u201d Kyungpook Math. J., vol. 57, no. 2, pp. 331\u2013360, 2017, issn: 1225-6951,0454-8124. doi: 10.5666\/KMJ.2017.57.2.331.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">22.\tV. Siwach and P. Madeti, \u201cAn unknotting sequence for torus knots,\u201d Topology Appl., vol. 196, pp. 668\u2013674, 2015, issn: 0166-8641,1879-3207. doi: 10.1016\/j.topol.2015.05.065.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">23.\tV. Siwach and P. Madeti, \u201cRegion unknotting number of 2-bridge knots,\u201d J. Knot Theory Ramifications, vol. 24, no. 11, pp. 1550053, 20, 2015, issn: 0218-2165,1793-6527. doi: 10.1142\/S0218216515500534.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">24.\tS. Vikash and M. Prabhakar, \u201cA sharp upper bound for region unknotting number of torus knots,\u201d J. Knot Theory Ramifications, vol. 22, no. 5, pp. 1350019, 21, 2013, issn: 0218-2165,1793-6527. doi: 10.1142\/S0218216513500193.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">25.\tP. Madeti and R. Mishra, \u201cMinimal degree sequence for torus knots of type (p, q),\u201d J. Knot Theory Ramifications, vol. 18, no. 4, pp. 485\u2013491, 2009, issn: 0218-2165,1793-6527. doi: 10.1142\/S021821650900704X.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">26.\tM. Prabhakar, \u201cA bound on the unknotting number,\u201d Int. J. Math. Anal., vol. 3, no. 5-8, pp. 339\u2013345, 2009, issn: 1312-8876,1314-7579. <\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">27.\tM. Prabhakar and R. Mishra, \u201cPolynomial representation for long knots,\u201d Int. J. Math. Anal., vol. 3, no. 5-8, pp. 325\u2013337, 2009, issn: 1312-8876,1314-7579.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">28.\tP. Madeti and R. Mishra, \u201cMinimal degree sequence for 2-bridge knots,\u201d Fund.   Math., vol. 190, pp. 191\u2013210, 2006, issn: 0016-2736,1730-6329. doi: 10.4064\/fm190-0-7<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">29.\tP. Madeti and R. Mishra, \u201cMinimal degree sequence for torus knots of type (p, 2p \u2212 1),\u201d J. Knot Theory Ramifications, vol. 15, no. 9, pp. 1141\u20131151, 2006, issn: 0218-2165,1793-6527. doi: 10.1142\/S021821650600497X<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">30.\tA. Sahore, K. Negi, A. S. Gill, M. Prabhakar, Classification of virtual links by arc shift move, arXiv:2502.08955, 2024. (submitted)<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">31.\tMohamad N. Nasser, Vaibhav Keshari, Madeti Prabhakar, Matrix representations of the twisted virtual braid group and its extensions, https:\/\/arxiv.org\/abs\/2506.03806, 2025 (submitted)<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">32.\tK. Negi, M. Prabhakar, The monoid structure of singular twisted virtual braids, arXiv:2502.08965, 2025. (submitted)<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">33.\tK. Kaur, A. Gill, and P. Madeti, \u201cArc shift number for some virtual knots,\u201d Trends in Mathematics, (DOI:10.1007\/978-981-13-5742-8_6), 2019<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">34.\tS. Vikash and M. Prabhakar, \u201cExact unknotting sequence of torus knots,\u201d in International Conference on Topology and Geometry, Japan-Mexico Topology Symposium, Japan, 2013, pp. 38\u201349.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">35.\tM. Prabhakar and M. Rama, \u201cPolynomial representation for links,\u201d in Proceedings of Knot Theory for Scientific Objects, Osaka, Japan, 2007, pp. 143\u2013154. 3 <\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">36.\tM. Prabhakar and M. Rama, \u201cA degree sequence for general knot,\u201d in Proceedings of Konan University, Konan University, 2005, pp. 169\u2013178. <\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t\t\t<li class=\"elementor-icon-list-item\">\n\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-icon\">\n\t\t\t\t\t\t\t<svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-chevron-circle-right\" viewBox=\"0 0 512 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M256 8c137 0 248 111 248 248S393 504 256 504 8 393 8 256 119 8 256 8zm113.9 231L234.4 103.5c-9.4-9.4-24.6-9.4-33.9 0l-17 17c-9.4 9.4-9.4 24.6 0 33.9L285.1 256 183.5 357.6c-9.4 9.4-9.4 24.6 0 33.9l17 17c9.4 9.4 24.6 9.4 33.9 0L369.9 273c9.4-9.4 9.4-24.6 0-34z\"><\/path><\/svg>\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-icon-list-text\">37.\tM. Prabhakar and M. Rama, \u201cMinimal degree sequence for torus knots,\u201d in Proceedings of Workshop on Topology of Knots, Nihon University, 2004, pp. 38\u201349.<\/span>\n\t\t\t\t\t\t\t\t\t<\/li>\n\t\t\t\t\t\t<\/ul>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div id=\"e-n-tab-content-1553712935\" role=\"tabpanel\" aria-labelledby=\"e-n-tab-title-1553712935\" data-tab-index=\"5\" style=\"--n-tabs-title-order: 5;\" class=\" elementor-element elementor-element-d9eaf2d e-con-full e-flex e-con e-child\" data-id=\"d9eaf2d\" data-element_type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Dr. M. Prabhkar Associate Professor Department of Mathematics Indian Institute of Technology Ropar Rupnagar, Punjab &#8211; 140001, India Office: C-M28, Mezzanine Floor, SAB building Phone: 0188123-2317 (O) Email : prabhakar@iitrpr.ac.in About Education Work Experience Publications Other Information Dr Prabhakar is an Associate Professor in the Department of Mathematics at IIT Ropar. After obtaining his doctoral [&hellip;]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_header_footer","format":"standard","meta":{"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":"","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-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-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-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-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-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-gradient":""}},"footnotes":""},"categories":[1],"tags":[],"class_list":["post-14128","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"\/maths\/index.php?rest_route=\/wp\/v2\/posts\/14128","targetHints":{"allow":["GET"]}}],"collection":[{"href":"\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"\/maths\/index.php?rest_route=\/wp\/v2\/users\/26"}],"replies":[{"embeddable":true,"href":"\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=14128"}],"version-history":[{"count":31,"href":"\/maths\/index.php?rest_route=\/wp\/v2\/posts\/14128\/revisions"}],"predecessor-version":[{"id":16836,"href":"\/maths\/index.php?rest_route=\/wp\/v2\/posts\/14128\/revisions\/16836"}],"wp:attachment":[{"href":"\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=14128"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=14128"},{"taxonomy":"post_tag","embeddable":true,"href":"\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=14128"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}