Course Description (view this short video description)
A mathematical knot is a closed loop in three-dimensional space. Many questions in knot theory are easy to state, but difficult to answer; we will set out to explore some of these questions. In this course, we will rigorously define knots and what it means for two knots to be equivalent. We will then discuss various mathematical techniques which arise in knot theory and apply these techniques to problems. The topics will include knot coloring, unknotting number, Alexander polynomials, surfaces whose boundary is a given knot, symmetries of knots, and higher dimensional knot theory, among others.
We will spend the first half of the term on so-called "classical" knot theory, building our foundation, and the second half of the term exploring various topics which are slightly more modern (the Jones polynomial, higher dimensional knot theory, the concordance group, etc.).
Along the way, we will touch on ideas and techniques from various areas of mathematics: topology, graph theory, linear algebra, abstract algebra, and number theory.
Prerequisites: MATH 8 or placement into Math 11
Textbook: Knot Theory by Charles Livingston
Remote Teaching (course structure and expectations):
Please read the consent to record page.
Because we are all in different time zones with different technology available to us, this course will be taught primarily asynchronously via pre-recorded video lectures, discussion boards, and daily class participation assignments. Synchronous portions of the course will generally be optional; these include office hours and problem sessions via Zoom. For extra practice, students should refer to the "Suggested Practice" column in the schedule of topics below.
Each week, a new module will appear in the Modules tab on Canvas. Each weekly module is broken up into 3 days of videos and activities. Students may complete the activities in any order, though the listed order is the intended progression. While I will not require that students complete each day's material on Monday, Wednesday, and Friday (although, if this is feasible, I encourage students to do so), I do expect that students interact with the course material at least 3 days per week.
Pre-recorded video lectures are hosted on TechSmith Relay. Students will need to log in with their Dartmouth ID to access these videos.
Discussion boards are located on Canvas under the "Discussions" tab. There will be a "General" discussion board for questions and conversations about the course, a "Random" discussion board for classroom chatter, a discussion board for each homework assignment, and several discussion boards related to class participation assignments (see the section on class participation below). Please note, while questions related to assignments are welcome on discussion boards, it is a violation of the honor code to provide explicit answers to homework problems on discussion boards. Each discussion board will come with instructions for proper usage.
Because this course is being taught remotely, and we will not have the opportunity to interact with one another in person, it is worth noting some online etiquette. There are two points I want to emphasize:
- Remember the human. Remember that your instructor and your peers are human. It is important to communicate with respect and understanding, especially in these stressful times. We will all make mistakes, we will all struggle with this new format. What's important is how we handle ourselves in the wake of our mistakes and struggles. A general guideline: if you wouldn't say it in person, don't say it online.
- You can't over-communicate. Without face-to-face communication, much of tone and implication gets lost. So use more words than necessary. Over-explain yourself. Re-read your messages and posts and think of the perspective of your peers.
The course grade will be based upon class participation and the scores on the midterm exam, homework, and the final project as follows:
To receive a score of "Credit" in this course, you must satisfy all of the following criteria:
- Receive a course grade of 60% or higher.
- Receive a score of 60% or higher in at least 3 of the 4 categories: class participation, weekly homework, midterm exam, and final project.
- Receive a score of 40% or higher in all categories.
Course grades will be tracked throughout the term. Students who show exceptional effort or distinguished work may be awarded with a citation at the end of the term.
In a face-to-face course, class participation points would be based on attendance, participation in discussions, actively engaging in group work, etc. In a remote course, we don't have many of these options. So our class participation grade will primarily be based on small, low-stakes assignments for each day. The goal of class participation assignments is for you to receive feedback throughout the week, and for me to keep track of how everyone is doing. These may take many forms including uploading a solution to problem, a 3-question quiz, or a post and response on a discussion board. Each class participation assignment is:
- marked as "CP" in the module task list.
- worth 3 points (final CP scores will be calculated as number of points earned out of points available),
- open for unlimited retries until the due date (generally the following Saturday at 11:59pm EDT),
Late Class Participation assignments will not be accepted. At the end of the term, up to 6 class participation points may be dropped.
Written homework assignments will be assigned weekly. They are due each Wednesday at 11:59pm EDT. Homework assignments will typically cover the material up through the previous Friday. So the first written assignment (soon to be available in the Week 1 module) covers the material from the Week 1 module and is due on Wednesday of week 2. Homework submissions will be via Canvas; more details to come.
Homework has two major purposes in this course:
- To keep everyone on track towards our learning goals.
- To give everyone an opportunity for creative thinking and problem solving.
Because of this, a typical homework assignment will contain about 60-70% direct applications and examples of the week's material with the remaining amount going towards problems that encourage you to think outside the box or discover something new.
No homework assignments will be dropped at the end of the term. However, each student can request a 2-day homework extension once during the term (no questions asked). To request an extension for a homework assignment, the student must contact the instructor before the assignment is due.
This course will have a midterm exam at around the halfway point. This will come in the form of a take-home, open-book (but not open-internet) exam. The purpose of this exam is to solidify our foundations in classical knot theory before continuing on to more exploratory topics.
During the second half of the term, each student will be expected to complete a final project exploring a related topic of their choosing. More details to come.
The Honor Principle:
Academic integrity is at the core of our mission as mathematicians and educators, and we take it very seriously. We also believe in working and learning together.
Collaboration on homework is permitted and encouraged, but obviously it is a violation of the honor code for someone to provide the answers for you.
On written homework, you are encouraged to work together, and you may get help from others, but you must write up the answers yourself. If you are part of a group of students that produces an answer to a problem, you cannot then copy that group answer. You must write up the answer individually, in your own words.
On the midterm exam, you may not give or receive help from anyone.
Accessibility Services etc.
Students requesting disability-related accommodations and services for this course are encouraged to schedule a phone/video meeting with me as early in the term as possible. This conversation will help to establish what supports are built into my online course. In order for accommodations to be authorized, students are required to consult with Student Accessibility Services (SAS; email@example.com; SAS website; 603-646-9900) and to email me their SAS accommodation form. We will then work together with SAS if accommodations need to be modified based on the online learning environment. If students have questions about whether they are eligible for accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential.
The academic environment at Dartmouth is challenging, our terms are intensive, and classes are not the only demanding part of your life. There are a number of resources available to you on campus to support your wellness, including your undergraduate dean (http://www.dartmouth.edu/~upperde), Counseling and Human Development (http://www.dartmouth.edu/~chd/), and the Student Wellness Center (http://www.dartmouth.edu/~healthed/).
Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with your instructor before the end of the second week of the term to discuss appropriate accommodations.
The syllabus page shows a table-oriented view of the course schedule, and the basics of course grading. You can add any other comments, notes, or thoughts you have about the course structure, course policies or anything else.
To add some comments, click the "Edit" link at the top.