Knot Theory - Mathematics 395 - Fall 2010

Homework

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Homework  Syllabus


HW #    Due Date    Read Do
Final Exam Th 12/16
Time: 9:30

The final exam will be cumulative, with some emphasis on material we learned after the last midterm, and on problems assigned for "team work". The exam will be roughly twice as long as one midterm (designed for two hours) but you'll have the full three hours available.
Extra office hours: W 12/15: 1:00-4:30.

Request items to be included on the "final exam notes".

 More review problems for the final (don't turn in): see this list.
30 M 12/6   Do two problems with your teammate from this list. Your new teammate is indicated on the problem list.
29 F 12/3   (i) Do these problems.
Also do two problems with your teammate from this list. Your new teammate is indicated on the problem list.
28 W 12/1   (i) True or false: The triangle-Y move preserves IKness. If true, give a detailed proof. If false, explain exactly where we would fail if we tried to mimic the proof that ILness is preserved by the triangle-Y move.
(ii) Same question as above, but for the Y-triangle move.
Also do two problems with your teammate from this list. Your new teammate is indicated on the problem list. You should try all these problems "for yourself" to make sure you can do them.
27 M 11/29   Do TWO problems with your teammate from this list. Your new teammate is indicated on the problem list.
26 F 11/19
M 11/22 		  Suppose G' is obtained by a triangle-Y move on a graph G. Prove that if G is intrinsically linked, then so is G'. (Hint: follow an outline similar to that of problem 8.4.)
Also do one problem with your teammate from this list (I already emailed you instructions.)
Mid 3 W 11/17 Covers HWs 15-25 24.  
25 F 11/12
M 11/15		 Sec 8.1 p. 219-222. 8.3-8.7
24 W 11/10 Sec 6.4 p. 175-176.
Sec 8.1 p. 215-218.		 6.22, 8.1, 8.2. Also do the following:
(i) Why does the book use the X polynomial instead of the bracket polynomial to show that the trefoil is not amphicheiral?
(ii) Is the Jones polynomial of an amphicheiral knot necessarily palindromic? Prove your answer.
(iii) Does amphicheiral imply invertibe? What about the converse? Prove your answers.
23 M 11/8 Sec 6.3 p. 165-167. 6.14-6.16.
22 F 11/5 Sec 6.1 p. 153-155. 6.5-6.8.
Hints for 6.6: First think about the bracket polynomial: (i) What is the bracket polynomial of the 2-component trivial link? (ii) Use induction on the number of crossings in L.
21 W 11/3 Sec 6.1 p. 147-152. 6.1-6.4.
20 M 11/1 Finish Sec 5.4. 5.27-5.29. Also do the following:
(i) Let K be a knot in closed braid form, with c crossings. Suppose we "thicken" K to a "flat band", homeomorphic to an annulus, still in the same closed braid form. Let L be the 2-component link consisting of the two boundary components of this band. Prove that lk(L) has the same parity as c.
19 F 10/29 Sec 5.4 p. 133-135. 5.21-5.26.
18 W 10/27 Sec 5.4 p. 127-132 but may skip 130-131.. 5.16-5.20. For 5.16, do only the first knot in the figure. Also do:
(i) Let K_1, K_2, ... be knots. True or false:  If u(K_n) --> infinity as n --> infinity, then c(K_n) --> infinity?  What about the converse? Prove your answers. (Recall: u = unknotting #, c = crossing #.)
17 M 10/25 Sec 5.2.

5.13, 5.14. Also do the following:
(i) Let col(K) = the least n >=3 such that the knot K is n-colorable. Suppose K_1, K_2, ... is a sequence of knots such that col(K_i) tends to infinity as i tends to infinity. Prove that c(K_i) tends to infinity (where c(K_i) denotes the crossing number of K_i).
Mid 2 F 10/22 Will cover HWs 6-15 14 and their corresponding sections.

 
16 W 10/18 Sec 5.1: p. 112-115.

5.5, 5.8, 5.10-5.12. Plus:
In the first theorem on p.100, why doesn't it say "reduced alternating projection"? Is it a typo? Or is it not necessary to say "reduced"?
15 F 10/15
W 10/8		 Sec 5.1: p. 107-112. 5.1-5.4, plus the following:
(i) How can we determine the number of components in a (p,q)-torus link without actually drawing the link? For example, how many components does the (12,15)-torus link have?
(ii) What kind of a knot is each component of a (p,q)-torus link?
(iii) If a (p,q)-torus link contains more than one component, what is the linking number between each pair of its components?
(iv) Does there exist a homeomorphism from the torus to itself that switches a meridian with a longitude? A proof is not necessary; but explain briefly how you arrive at your answer.
14 W 10/13 Sec 4.3: read p.100, skip proof of 2nd theorem on p.99; read bottom of p.104 to top of p.106. 4.27-4.29.
13 M 10/11 Sec 4.3 p.95-99. 4.18-4.22. Also do the following:
Draw diagrams to show the surface in Figure 4.51 is homeomorphic to a torus minus a disk.
12 F 10/8 Sec 4.2. 4.13-4.17.
11 W 10/6 Sec 4.1: p. 77-83.
4.4-4.6, 4.8-4.10. (In 4.10 you don't have to use Euler characteristic; use any method you like.)
10 M 10/4 Sec 4.1: p. 73-76. 3.15, 4.3.
There is a mistake in Figure 4.13; what is it?
Also do problems 1.2, 1.3 from Promolov's Intuitive Topology.
9 F 10/1 Sec 3.3.
Sec 4.1: p. 71-72.		 3.14, 4.1, 4.2.
8 W 9/29 Sec 3.2 p. 64-65. 3.4, 3.7, 3.9, 3.10, 3.11, 3.12a. Also do the following problem:
A. True or false: For every knot K, there is a minimal-crossing diagram of K such that changing some set of u(K) crossings in that diagram gives an unknot. Does the book give any evidence for or against this statement?
7 F 9/24
M 9/27		 Sec 3.1. 2.27, 2.28, 2.29, 2.31.
3.1-3.3.
Midterm-1(with pics)

Mid-1 w/o pics

 W 9/22 Will cover HWs 1-5 and their corresponding sections.
The best way to prepare  for the midterm is to make sure you can do every homework problem without looking at the book or your notes.		  
6 M 9/20 Sec 2.4. (We may come back to Sec 2.3 later.) 2.2, 2.3, 2.5-2.9. Also do the following problems:
A. In Dowker notation, does every permutation of 2, 4, ..., 2n correspond to a knot?
B. On p. 38, what does "subpermutation" mean? Give a clear and detailed explanation.
C. In Figure 2.9, the two projections do not seem to be related by flipping crossings; why are they mirror images?
5 F 9/17 Section 2.1 and 2.2. 1.32-1.37.
4 W 9/15 Section 1.6. 1.18-1.31, except 1.24 & 1.30; 1.20 is optional.
Hint for 1.18: take two rubber bands and "hook" them together in such a way that they won't come apart by just pulling (even though they form a splittable link).
3 M 9/13 Sections 1.4 and 1.5. 1.10-1.17. For problem 1.11, you don't have to use Reidemeister moves; it's enough to draw a sequence of diagrams, as long as it is very clear how you go from each diagram to the next.
2 F 9/10 Section 1.3. 1.8, 1.9. Also do the following problems:
(i) Prove that the knots 5_1 and 5_2 (p. 280) are both invertible.
(ii) Is the knot 6_3 invertible? Prove your answer.
Diagrams for inverting 6_3.
1 W 9/8 Sections 1.1 & 1.2. [1.1], 1.2, [1.3], 1.4-1.7. Do not turn in the problems in square brackets.