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In knot theory, an area of mathematics, the link group of a link is an analog of the knot group of a knot. They were described by John Milnor in his Bachelor's thesis, (Milnor 1954).

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PLATFORM LOGIN To access your user account on the Distance Learning system, you need to enter your login information into the login field. You will receive your login information via email after enrolment. After entering your username and password, you will be able to access the Distance Learning platform and start your first courses. When you log into you account, you will see different tabs with numerous functions available on the DL platform. You can see the User Service menu, the Learning Support menu, documents and contact information. On the left side of the screen is the system of additional options, which was developed to give you better insight into your upcoming obligations and facilitate the use of the Distance Learning platform. As you can see, it consists of several blocks. LIVE CLASS The Live Class block shows the timetable and links for accessing online lectures, if any such lecture has been scheduled as part of your courses. DASHBOARD Dashboard is the panel with options that are used frequently in the Distance Learning system. Here you can quickly access new messages through Messages, review yourchat consultations with the Chat option, see the number of currently active users on the platform and the number of active users in the last 24 hours in the Online section, as well as access many other services that may prove useful for learning, grouped in the Other Services section. You can configure the Dashboard the way it suits you, change the position of icons, remove the ones you do not need, and move the ones you frequently use to the top. In addition, every block in the side bar can be set up individually, or you can remove blocks by clicking on the x in the upper right corner. If you remove a block by mistake, you can restore it by going to the Settings option at the bottom of the screen. ONLINE SUPPORT The Online Support block is the option that enables you to communicate with the technical support personnel via chat. When the support status is online, the support window will open, where you can ask all kinds of technical questions. In addition, you can also contact technical support via email. You will find technical support in the Contact Information menu. CALENDAR The Calendar option represents the general calendar of recommended activities in all courses that have been assigned to you. The Calendar enables you to see your upcoming commitments, such as completing module tests, or submitting assignments and tasks. By clicking on a day in the Calendar, you can add a reminder for that day or postpone a commitment. These are recommended dates, which means that the student is not required to do the test or complete the assignment on that specific day, but that he can fulfil his obligations as long as the course is active. However, following the recommended dynamics will enable you to complete your training within the specified time. DICTIONARY Here you can find our English Dictionary, a function that allows you to use the English-Serbian and Serbian-English dictionary at any time. You only need to type the word that you wish to translate to English or Serbian, and the translation will appear below. SEARCH BAR The Search Bar option is located in the top right corner of the platform. Wherever you are on the platform, you can search for terms that you need, using our advanced Search Bar. This option searches for the desired term in all active courses, including units and the content of all courses. After entering the word and pressing the magnifier button, magnifier button, you will get a list of all courses and lessons in which the search term appears. COURSES All of your currently active courses can be found in the central section of the platform. There are two columns next to the name of the course. The first represents the percentage of covered material, whereas the second column shows the credits or points percentage. The green slider is for active, and the grey one for completed courses. You will sometimes see an asterisk or an exclamation mark next to the name of the course. If a course expires in less than ten days, the exclamation mark will appear next to its name, as a warning. If it is a bonus course that you have been awarded, then you do not have to complete it unless you want to. Bonus courses have an asterisk next to their name. If you position the cursor over the course name, it will open a window with a description of every available course. The same window displays the number of times that you accessed the course, as well as the date of assignment and expiration of the course. You will find shortcuts to recently opened lessons under the list of assigned courses. Here you can see the dates when you last opened lessons, frequency of opening lessons, as well as the names of the courses and teaching modules and their accompanying lessons. Pausing a Course In the bottom right corner of every course is the Pause Course option. This option allows you to pause any DL course that you have been assigned. The course will remain inactive until you resume it. You can resume the course whenever you want. This way, you will pause the time specified for the completion of the course. When you resume the course, you will have the same number of remaining days at your disposal as when you paused the course. It is important to know that even when you pause a course, you are still able to access Live Stream and Chat consultations, but not materials from the course. DL courses can be paused and activated as many times as you want during the school year. Exactly one year after your enrolment, all paused courses will be automatically activated. When the course is activated, you will receive an email notification. COURSE CONTENT Courses are opened by clicking on their name. When you click on a selected course, you will see a page with basic information about the course. You will see the number of credits that the course brings and its expiration date at the beginning. Below that, you can see the estimated time for completing the course. It is important to note that this is just an indicative estimate and that you have the freedom to adapt your learning and coverage of the material to your needs. Below the estimated time, you will find a short description of the course, which summarises the content and structure of the course. The objective of the course will introduce you to the outcomes of the course that you are attending, in other words, specific things that you will learn by covering the material. MODULES The next tab is Modules and Units, which contains all modules and units that make up a course. Modules represent more general units that contain individual teaching units. A module includes a summary, a module test for which you get credits necessary for completing the course, and occasionally, practical exercises for knowledge assessment that include assignments and tasks, as well as additional material. Additional materials within a module are not compulsory, but it is certainly advisable to read them in order to master the material more successfully. LITERATURE At the end of the module, you will find the list of recommended literature and useful links that you can use if you want to learn more about the topic of the course. We should underline that these books are not compulsory for the successful completion of the course. Some courses also have information on the software required to successfully complete the course, under the list of recommended literature. We should also mention ebrary, the online library available in the Learning Support menu. Ebrary contains a number of useful sources from the fields of IT and business. This service is very easy to navigate. To access library resources, click on the active link that will lead you directly to ebrary, without having to create a new user account. When you open the homepage, you will see the tutorial that explains how to use the library and the browser, and then the classification of library titles by publishers and areas. Using the ebrary service is not compulsory, but it is advisable, because this service was provided to students to additionally improve their DL platform learning process. EduWall Next to the Modules and Units tab, you can see the tab for accessing the EduWall, but only in courses that have been assigned a wall. EduWall is a place where students can communicate and exchange information. It is quite reminiscent of Facebook, except that the primary purpose of EduWall is educational. Students can discuss lesson-related topics with their fellow students and instructors on the wall, share and download useful links, pictures, documents and files associated with the course, solve tasks and problems uploaded by instructors. Another way to access EduWall is to go to the Learning Support menu and select the option EduWall. EduWall is an additional activity, which means that you do not have to be active there, although it is recommended, for the purpose of more efficient and active learning. VIDEO ARCHIVE and LIVE CLASS Video Archive contains all video lectures available on Live Stream in the Live Class menu. If you are unable to watch the lecture in real-time, all video lessons are available in the Video Archive. You can watch all videos from our Video Archive whenever you want. TEACHING UNIT Now that we have explained the structure of the course, we will move on to the structure od a teaching unit. You will see the name and number of the lesson at the beginning. You can see that there is an eye icon next to the name of the lesson, which is initially grey. When you read the unit and when the unit timer reaches zero, the eye icon will change colour, so you would know which lessons you have read. To access the content of a lesson, click on its name. Just below the name of the lesson there is a timer on the right which counts down while you are reading, using the formula of 900 characters per minute. The lesson will be regarded as completed only after the specified time has expired, in other words, the percentage of covered material for the specific lesson will be recorded, while, of course, you read the lesson for as long as you like. Every lesson begins with a summary. Summary is a short overview of the lesson and it includes its most important aspects. Same as the module summary, lesson summary can be read before you start reading the lesson, but afterwards as well, so you would remind yourself of the crucial parts of the teaching unit. GLOSSARY The textual content of the lesson is located below the summary. You will notice that some words in the lesson are underlined. If you click on them, you will see an explanation, i.e. glossary. This way, you can learn the meaning of all unfamiliar words, improving your understanding of the material. In addition to detailed textual material, lessons also include a variety of pictures and examples that provide better understanding and practical examples for the content learned. You are not obligated to answer the question, but we advise you to utilise this function in order to improve your learning method. After answering the question, you will be given the correct answer and its explanation. LESSON ELEMENTS Since we explained the textual part of the lesson, it is time to show you the other lesson elements. In addition to the textual part, lessons on the DL platform also include multimedia, progress tests and PDF material located on the left. Multimedia Multimedia contain numerous examples, and present lesson content in a dynamic and interesting way. We should note that multimedia is a compulsory part of the course, so you need to watch this content in order to successfully complete the course. When you click on Multimedia, it will open the video in a new window. The Player has the following options: play, pause, rewind(10 seconds), autoplay, speed up and slow down multimedia, sound control, full screen mode, etc. Multimedia for the course is an integral part of the course and it should not be confused with traditional videos with recorded lessons that can be found in the Video Archive or Live Stream. Multimedia related to the lesson covers the same subject as the lesson itself and represents a compulsory learning material. Videos from the Video Archive can help you master the lessons, but are not compulsory. The status of the video material from the Video Archive is defined by the rules of your educational institution, so it is important to be informed about it. PDF In addition to the multimedia, another lesson element is the PDF material which is identical to the lesson material and can be downloaded to your computer, as long as the course is active. PROGRESS TESTS With regard to lesson elements, we should also mention progress tests. Progress tests are exercise tests with which you can test your knowledge of the lesson. You can do these tests whenever you want, because they are just for practice and their results will not be recorded anywhere, nor will they affect your achievement in the course in any way. When you start a progress test, you will see the available time for completing the test and the number of questions in the top right corner. Questions on these tests can be different: questions with several predefined answers, linking questions, questions where you need to insert or write down the answer, as well as questions with images and video files. When you answer all questions, click on the Finish Test button and you will see your results and explanations of correct answers. Module Test As we mentioned earlier, at the end of every module, you will find a module test the purpose of which is to test your knowledge of that module. Unlike progress tests that serve as an exercise and can be done as many times as you want, module tests can be done only once, because points scored on this test will be included into your achievement in the course. Module tests cover the material from the entire module, and the number of questions varies, depending on the total number of lessons within the module. Types of questions are the same in module tests and progress tests. When you answer all questions, click on the Finish Test button. After you submit your answers, click on the See Result button to see a detailed review of the correct answers and their explanations. You can also view the answers and their explanations within the Learning Support service, Test Review, See Result. Tasks and Assignments In addition to module tests, the knowledge assessment system on the DL platform includes tasks and assignments. If a certain module contains tasks, they will be located under the module test. These tasks represent a practical verification of the knowledge you acquired in the module, and will help you transform theoretical into practical knowledge. There are two forms of knowledge assessment - tasks and assignments. These two forms are based on the same principle, but they differ in visibility after the approval of the course instructor. Your task will become visible to other students who successfully completed the same tasks, whereas the assignment is visible only to you. By clicking on the name of the task or assignment, you will see a description of that task or assignment in a new window. When sending the completed project to the course instructor, you can enter a short text or a comment about the assignment. When you enter the text of the comment, click on the Save Changes button and then on Send Assignment. This way, you will submit your assignment to the course instructor, who can then decide to return it for revision or accept it, awarding you with stars that represent the number of points that the assignment carries. Assignment and File Review You can quickly and easily find all assignments and files that you sent to the course instructor in the Learning Support menu, in Assignment Review and File Review tabs. Communication with the Course Instructor If you have the support of the course instructor for the given course, after he or she assesses your work, you will receive a notification, which can be seen in the Personal Messages section. Every course and every lesson contains a link Ask Instructor, where you can ask questions about the given course. If the course has several instructors, the question will be forwarded to the lead instructor. When you click on the link, it will open a new window where you can enter your question. After you finish, click on the Send Message button and your question will be sent. When the course instructor responds, you will find the message in the Personal Messages option, in the User Service menu. Chat Sessions Another form of communication with the course instructor are chat consultations. Users of the DL platform can use chat consultations, i.e. conversation in real-time with the instructor and other participants in the DL process. After the course instructor schedules consultations, students will receive a notification message with the date and time of the consultations. Chat sessions can be opened from the Dashboard, where you can see the name of the chat consultation and the time when it begins. By clicking on the name of the Chat Consultation, you will open the chat window. In addition, chat consultations can also be started with the Chat option, in the User Services menu, which will open a window with the list of all available chat sessions. You can see the name, associated course, status and start date of every consultation. You can join some chat sessions by clicking on the link. Another thing that you view here are transcripts of all completed chat sessions. User Services Expired courses Let's return to assigned courses. We mentioned earlier that each course has a specified duration, in other words, a start date and an expiration date. You can see these dates when you move the cursor over the name of the course. When a course expires, it will no longer be available in the list of assigned courses, but in the list of expired courses located within the User Services menu. Courses that have expired can be complete or incomplete. You can tell if the course is complete or not based on the indicator next to it. In addition to the list of expired courses, you will also find information about covered material and credits achieved here. Course Extension If the student fails to fulfil all the requirements and the course expires, the specific course will be moved to the list of expired courses. If the Extend Course button is available, you can extend the course for free by clicking on it. If the course has the option of free extension, the extension will include additional 30% of the total duration of the course. Additional extensions are possible, but will be additionally charged in accordance with our price list of services. Remaining Requirements The Remaining Requirements option, in the User Services menu, will help you check if you have any remaining requirements in expired courses. Here you will find a list of courses that have expired, but remain incomplete. By clicking on the desired course, you will open a complete list of requirements for that course. SCHEDULE Another option you can use to keep track of your DL courses is the Schedule option, which, if defined by your training regulations, is available in the Learning Support menu. Schedule provides a very clear and organised plan of DL course assignment: compulsory courses, bonus courses and English Language courses. If this option is not available on the platform, you will receive your schedule via email, or in the Documents menu - under Instructions. Personal Record You can enter your personal information with the Personal Record option in the User Services menu. Here the user can edit his personal information and password, upload a picture, configure email settings, find out which social networks he used to log in, etc. Achievement Students can keep track of their achievement with the User Service menu and Achievement option, where they can see a detailed review of covered material and achieved credits for each assigned course and each module within the course. Statistics Students can use the Statistics option to see their course results, a detailed review of the time spent on the system, points scored in tests, and results achieved in assignments and tasks for each individual course. LEARNING JOURNAL This option is available within the User Service menu and allows you to follow your learning progress with the help of tables and charts. When you select an active course, you will see a table with recommended dates for each requirement. You are able to plan the completion of each requirement the way it suits you. You can leave a comment or a reminder for each date, in order to keep track of the fulfilment of your plans and your learning progress. In addition to the tables, the learning journal also allows you to follow your progress in courses using individual and cumulative graphs. Individual graphs show the recommended progress rate and your actual progress rate, whereas cumulative graphs give insight into recommended requirements that you have completed and the delay in their fulfilment. FINANCES Since we are still talking about the User Service menu, we should mention the Finances option, which enables you to see an overview of your finances. DOCUMENTS Finally, we should mention DL documents and the related options. CERTIFICATION If the Certification option is available at the institution of your choice, it will lead to information about the types of certificates available in different study programs, as well as the requirements for obtaining them. INSTRUCTIONS The Documents menu also contains the Instructions option, where you can obtain different instructions and tutorials on the operation of the DL platform and questions about the training regulations, task and assignment completion and submission, and effective learning techniques. This is only a part of what distance learning on the LINK group Distance Learning system offers. There are many other options available on the DL platform, but you will need to discover them for yourself in the process of learning and exploration. We hope that you will join us in the new school year. If you have any questions, our learning consultants will be glad if you get in touch.



The Whitehead link is link homotopic to the unlink, but not isotopic to the unlink.
The Whitehead link is link homotopic to the unlink, but not isotopic to the unlink.

The link group of an n-component link is essentially the set of (n + 1)-component links extending this link, up to link homotopy. In other words, each component of the extended link is allowed to move through regular homotopy (homotopy through immersions), knotting or unknotting itself, but is not allowed to move through other component. This is a weaker condition than isotopy: for example, the Whitehead link has linking number 0, and thus is link homotopic to the unlink, but it is not isotopic to the unlink.

The link group is not the fundamental group of the link complement, since the components of the link are allowed to move through themselves, though not each other, but thus is a quotient group of the link complement's fundamental group, since one can start with elements of the fundamental group, and then by knotting or unknotting components, some of these elements may become equivalent to each other.


The link group of the n-component unlink is the free group on n generators, , as the link group of a single link is the knot group of the unknot, which is the integers, and the link group of an unlinked union is the free product of the link groups of the components.

The link group of the Hopf link is  Z 2 . {\displaystyle \mathbf {Z} ^{2}.}
The link group of the Hopf link is

The link group of the Hopf link, the simplest non-trivial link – two circles, linked once – is the free abelian group on two generators, Note that the link group of two unlinked circles is the free nonabelian group on two generators, of which the free abelian group on two generators is a quotient. In this case the link group is the fundamental group of the link complement, as the link complement deformation retracts onto a torus.

The Whitehead link is link homotopic to the unlink – though it is not isotopic to the unlink – and thus has link group the free group on two generators.

Milnor invariants

Milnor defined invariants of a link (functions on the link group) in (Milnor 1954), using the character which have thus come to be called "Milnor's μ-bar invariants", or simply the "Milnor invariants". For each k, there is an k-ary function which defines invariants according to which k of the links one selects, in which order.

Milnor's invariants can be related to Massey products on the link complement (the complement of the link); this was suggested in (Stallings 1965), and made precise in (Turaev 1976) and (Porter 1980).

As with Massey products, the Milnor invariants of length k + 1 are defined if all Milnor invariants of length less than or equal to k vanish. The first (2-fold) Milnor invariant is simply the linking number (just as the 2-fold Massey product is the cup product, which is dual to intersection), while the 3-fold Milnor invariant measures whether 3 pairwise unlinked circles are Borromean rings, and if so, in some sense, how many times (that is to say, the Borromean rings have a Milnor 3-fold invariant of 1 or –1, depending on order, but other 3-element links can have an invariant of 2 or more, just as linking numbers can be greater than 1).

Another definition is the following: let's consider a link . Suppose that for and . Pick any Seifert surfaces for the respective link components, say, , such that for all . Then the Milnor 3-fold invariant equals minus the number of intersection points in counting with signs; (Cochran 1990).

Milnor invariants can also be defined if the lower order invariants do not vanish, but then there is an indeterminacy, which depends on the values of the lower order invariants. This indeterminacy can be understood geometrically as the indeterminacy in expressing a link as a closed string link, as discussed below (it can also be seen algebraically as the indeterminacy of Massey products if lower order Massey products do not vanish).

Milnor invariants can be considered as invariants of string links, in which case they are universally defined, and the indeterminacy of the Milnor invariant of a link is precisely due to the multiple ways that a given links can be cut into a string link; this allows the classification of links up to link homotopy, as in (Habegger & Lin 1990). Viewed from this point of view, Milnor invariants are finite type invariants, and in fact they (and their products) are the only rational finite type concordance invariants of string links; (Habegger & Masbaum 2000).

The number of linearly independent Milnor invariants of length for m-component links is , where is the number of basic commutators of length k in the free Lie algebra on m generators, namely:


where is the Möbius function; see for instance (Orr 1989). This number grows on the order of .


Link groups can be used to classify Brunnian links.

See also


  • Cochran, Tim D. (1990), "Derivatives of links: Milnor's concordance invariants and Massey's Products", Memoirs of the American Mathematical Society, American Mathematical Society, 427
  • Habegger, Nathan; Lin, Xiao Song (1990), "The classification of links up to homotopy", Journal of the American Mathematical Society, 2, American Mathematical Society, 3 (2): 389–419, doi:10.2307/1990959, JSTOR 1990959
  • Habegger, Nathan; Masbaum, Gregor (2000), "The Kontsevich integral and Milnor's invariants", Topology, 39 (6): 1253–1289, doi:10.1016/S0040-9383(99)00041-5, preprint.
  • Milnor, John (March 1954), "Link Groups", Annals of Mathematics, Annals of Mathematics, 59 (2): 177–195, doi:10.2307/1969685, JSTOR 1969685
  • Orr, Kent E. (1989), "Homotopy invariants of links", Inventiones Mathematicae, 95 (2): 379–394, doi:10.1007/BF01393902
  • Porter, Richard D. (1980), "Milnor's μ-invariants and Massey products", Transactions of the American Mathematical Society, American Mathematical Society, 257 (1): 39–71, doi:10.2307/1998124, JSTOR 1998124
  • Stallings, John R. (1965), "Homology and central series of groups", Journal of Algebra, 2 (2): 170–181, doi:10.1016/0021-8693(65)90017-7
  • Turaev, Vladimir G. (1976), "The Milnor invariants and Massey products", Studies in Topology-II, 66: 189–203
This page was last edited on 20 April 2018, at 07:55
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