Mathematics 148

Knot Theory

Pitzer College, Fall 2012

Homework Assignments

I need to know how much time the weekly assignment is taking, so let me know! If you hit 8 hours, I would encourage you to stop. (Of course if you are driven to do the problems, great!)

A note on numbering of the Exercises in Livingston's book. In general, let's number an exercise as n.m.l where n is the chapter, m is the section, and l is the exercise. This breaks down with Chapter 1, which has no sections.

HW 1 Due Monday, 9/10 1.1, 1.2, 1.4, 1.6 (I think you can find the answer to 1.6 on the internet. Please don't look!)

HW 2 Due Monday, 9/17 1. Show that any knot that lies in a plane is equivalent, by triangle moves, to a triangle. This can be restated as, Every planar knot is trivial. (See exercises 2.3.1, 2.3.2, 2.3.3.) There is a really hard way to do this and a MUCH simpler way to do it and I can give you a hint that should lead you to the simpler way. So, if you get stuck, ask for the hint.
Do also, 2.3.6, 2.4.4, 2.5.4, 2.5.5 (Try to do 2.5.5 for all knots in the table at the back of the book up to 7 crossings.)

HW 3 Due Monday,10/1 3.2.3, 3.2.5, 3.2.6, 3.2.8, 3.3.4, 3.3.5, 3.3.7, 3.4.2, 3.4.5, 3.4.6

HW 4 Due Wednesday,10/10 HW4

HW 5 Due Wednesday,10/31 4.2.4, 4.2.8, 4.2.9, 4.2.10, 4.3.1, 4.3.5
Notes: Exercise 4.2.10 refers to a surface "of the type illustrated in Figure 4.11." What this means is a disk with an arbitrary (but finite) number of pairs of bands attached like the first two pairs on the left, and then with an arbitrary (but finite) number of single bands attached like the two bands on the right. If the surface of Figure 4.11 is sitting in 3-space (embedded) as shown, then its boundary is an unlink. So the exercise is saying that every connected orientable surface with boundary is intrinsically the same as a surface like the one in Figure 4.11.

HW 6 Due Monday,11/19 HW 6

HW 7 Due Wednesday,11/28 HW 7

HW 8 Due Wednesday,12/5 HW 8

HW 9 Due Wednesday,12/12 HW 9

HW 1	Due Monday, 9/10	1.1, 1.2, 1.4, 1.6 (I think you can find the answer to 1.6 on the internet. Please don't look!)
HW 2	Due Monday, 9/17	1. Show that any knot that lies in a plane is equivalent, by triangle moves, to a triangle. This can be restated as, Every planar knot is trivial. (See exercises 2.3.1, 2.3.2, 2.3.3.) There is a really hard way to do this and a MUCH simpler way to do it and I can give you a hint that should lead you to the simpler way. So, if you get stuck, ask for the hint. Do also, 2.3.6, 2.4.4, 2.5.4, 2.5.5 (Try to do 2.5.5 for all knots in the table at the back of the book up to 7 crossings.)
HW 3	Due Monday,10/1	3.2.3, 3.2.5, 3.2.6, 3.2.8, 3.3.4, 3.3.5, 3.3.7, 3.4.2, 3.4.5, 3.4.6
HW 4	Due Wednesday,10/10	HW4
HW 5	Due Wednesday,10/31	4.2.4, 4.2.8, 4.2.9, 4.2.10, 4.3.1, 4.3.5 Notes: Exercise 4.2.10 refers to a surface "of the type illustrated in Figure 4.11." What this means is a disk with an arbitrary (but finite) number of pairs of bands attached like the first two pairs on the left, and then with an arbitrary (but finite) number of single bands attached like the two bands on the right. If the surface of Figure 4.11 is sitting in 3-space (embedded) as shown, then its boundary is an unlink. So the exercise is saying that every connected orientable surface with boundary is intrinsically the same as a surface like the one in Figure 4.11.
HW 6	Due Monday,11/19	HW 6
HW 7	Due Wednesday,11/28	HW 7
HW 8	Due Wednesday,12/5	HW 8
HW 9	Due Wednesday,12/12	HW 9