Knot Theory 581
Instructor
John Starrett,
835-5763, Weir 132,
jstarret@nmt.edu
Text: Knot Theory and its Applications, paperback
Author: Kunio Murasugi
Publisher: Birkhauser Boston (October 3, 2007)
ISBN-10: 081764718X
ISBN-13: 978-0817647186
| Homework | |
| Week 1 | 1. Show equivalence of fig. 8 knot and its mirror image by a sequence of R moves. 2. Prove linking number invariant under R moves Read Chapter 1 |
| Week 2 | 1. Prove tricolorability is invariant under R moves 2. Show fig. 8 not tricolorable. Read Chapters 2 and 3 |
| Week 3 | 1.
Show that by following a path through the knot and marking each
crossing with successive integers, you will always have pairs of even
and odd numbers associated to each crossing. 2. Find the relations induced by the Reidemeister moves on the extended Dowker-Thistlethwait notation |
| Week 4 | 1. Read through the Kauffman / Lambropoulou paper and mark the sections you don't understand. 2. Outline of the proof of the main theorem. State what each step is supposed to do on the way to the final result. |
| Week 5 | Read
Chapters 5 and 6 of the text. You may find it beneficial to skim at
first to see how the Alexander polynomial is developed, and then read
with more care a second and third time. Mark or note the parts you
still don't understand. 1. Find two different projections for a trefoil and draw two different Seifert surfaces for each. Do the same with the figure 8 knot. |
| Week 6 | Turn in the preliminary version of your final project, with abstract, outline and bibliography. |
| Week 7 | 1. Prove that cutting between arrow tips of an oriented regular knot diagram gives a collection of closed curves. 2. Prove that by connecting these curves with twisted bands you obtain an oriented surface. 3. Prove that the regular knot diagram can always be checkerboard colored. |
| Week 8 | Read chapter 6 and do exercises 6.1.1, 6.2.3, 6.4.1 and prove proposition 6.1.1 |
| Week 9 | Due Tues Nov 10 Significant update to paper. Must have bibliography, abstract, introduction, definitions, some illustrations (if any will be used in the final paper), and a portion of the body of the paper.For those of you who are using computer code, turn in some code or code output also. |
| Week 10 | |
| Week 11 | |
| Week 12 | |
| Week 13 | |
| Week 14 | |
| Week 15 |
Final projects from previous class
Final projects from other knot theory class