An Introduction to Math Beyond Calculus: Knot Theory

By studying the algebraic and geometric properties of knots in the three-sphere, including knot projections and Reidemeister moves, knot colorings, the Seifert matrix and the Alexander polynomial, knot signatures, the fundamental group, we will answer questions such as “What surfaces can a given knot bound”, or “How many times does a knot need to pass through itself to become unknotted”, or “What happens if I cut a knot and reglue the pieces in a different way?”. Depending on time, we may explore some more advanced topics, such as knot concordance, the relation between knots and 3-manifolds, and some more advanced algebraic invariants of knots.

Charles Livingston, *Knot Theory*

Instructor | Ina Petkova |
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Class | (11) MWF 11:30 - 12:35 |

X-hour | (11X) T 12:15 - 1:05 |

Classroom | Kemeny 201 |

Office | Kemeny 317 |

Office Hours | M 4 - 5pm W 4 - 5pm |

ina.petkova AT dartmouth.edu |

There will be an in-class midterm exam and a final exam. The exams are scheduled as follows:

Midterm | Wednesday, February 7 | in-class | |
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Final Exam |
Take home | Details TBA |

If you have a conflict with the midterm exam because of a religious observance, scheduled extracurricular activity such as a game or performance (not practice!), scheduled laboratory for another course, or similar commitment, please see your instructor at least one week in advance so possible alternative arrangements can be pursued.

All students must take the final at the scheduled time, unless they are scheduled by the registrar to have two conflicting examinations or three examinations on a single calendar day. In particular, no final will be given early or late to accommodate student travel plans. If you make travel plans that later turn out to conflict with the scheduled exam, then it is your responsibility to either reschedule your travel plans or take a zero in the final.

If you have a question about how your exam was graded, you can ask your instructor; to have your exam regraded, please submit your question in writing to your instructor.

- Homework will be assigned approximately once a week, and will be posted on the homework page.
- In homework (and on exams), be sure that you show your work, explain all steps, and write neatly. A correct answer with no work shown or that cannot be read will receive minimal credit. This is good practice for what will be expected on exams.
- If you have a question about how homework was graded, you can ask your instructor; to have it regraded, please submit your question in writing to your instructor.
- No late homework will be accepted.

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.

Cooperation on homework is permitted and encouraged, but if you work together, try not take any paper away with you—in other words, you can share your thoughts (say on a blackboard), but try to walk away with only your understanding. In particular, you must write the solution up individually, in your own words. This applies to working with tutors as well: students are welcome to take notes when working with tutors on general principles and techniques and on other example problems, but must work on the assigned homework problems on their own. Please acknowledge any collaborators at the beginning of each assignment.

On exams, you may not give or receive help from anyone. Exams in this course are closed book, and no notes, calculators, or other electronic devices are permitted.

Plagiarism, collusion, or other violations of the Academic Honor Principle will be referred to the Committee on Standards.

The course grade will be based upon the scores on the midterm exam, homework, and the final exam as follows:

Class attendance and participation | 10% |

Homework | 30% |

Midterm | 30% |

Final Exam | 30% |

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.

Students with disabilities who may need disability-related academic adjustments and services for this course are encouraged to see their instructor privately as early in the term as possible. Students requiring disability-related academic adjustments and services must consult the Student Accessibility Services office (Carson Suite 125, 646-9900, Student.Accessibility.Services@Dartmouth.edu). Once SAS has authorized services, students must show the originally signed SAS Services and Consent Form and/or a letter on SAS letterhead to their professor. As a first step, if students have questions about whether they qualify to receive academic adjustments and services, they should contact the SAS office. All inquiries and discussions will remain confidential.