- Mike Landis' presentation
- Dan Barnak's presentation
- Olivia Dewey's presentation

**Homework**:

- To turn in on Thursday:
- 6.22
- Amphicheirality—look up the HOMFLY polynomial of your pet knot (note: their "z" is our "m" and their "a" is our "l") and use this to determine whether your knot is amphicheiral (see exercise 6.26). Note you may take for granted that the HOMFLY polynomial can detect both amphicheiral and non-amphicheiral 7-crossing knots (I'll turn this back to you on Friday so you can add it to the wiki)

- Wiki—the finished wiki project is due at the time our scheduled final exam would end:
**Tuesday, May 5, 5:00**. Still to do:- Final cleanup (look at the talk page and fix all outstanding issues)
- Braid representation: you did this on a previous homework. Now transfer it to the wiki. Draw your knot as a closed braid (in the justification box). Give the corresponding braid representation (on the main screen, outside the justification box)
- Span of the bracket polynomial: add a new section for this under genus. In the justification box you may use the lemma on page 160—you do not have to give the longer justification that you did in the homework. (This document should be self contained, so in your justification don't refer to "the lemma on page 160." Write a complete justification.)
- Just like we did for the Jones polynomial, look up the Alexander polynomial and the HOMFLY polynomial in the knot table. Then add them to your wiki page.
- Amphicheirality—I'll turn this back to you on Friday so you can add it to the wiki

**Note**: if you add this new material (braids, span, Alexander polynomial, HOMFLY polynomial) to the wiki by Thursday, then I will be able to look it over and comment on it so that you can clean it up by the final due date.

- Amphicheiral knots
- The bracket, X, and HOMFLY polynomials for amphicheiral knots

**Homework**:

- 6.22 (due next Thursday)
- Begin cleaning up your wiki—go to the talk page and make the changes mentioned there.

**Presentations**:

- Monday: Mike, Dan, Olivia
- Thursday: Richard, Tan, Topher, Erik

- Alexander polynomial
- HOMFLY polynomial

**Homework**:

- 6.14 (you may use the resolving tree method I showed you in class), 6.15, compute the HOMFLY polynomial for the figure eight knot (you may use the resolving tree method I showed you in class), 6.21

**Read for next class**:

- 6.4

**Announcements**:

- Email me if you have a preference of presenting on Monday versus Thursday next week (first come first served for time slots)

- Alternating projections and crossing number
- The Alexander polynomial
- resolving trees

**Homework**:

- Wiki:
- Fill in the section on the genus of your pet knot
- Look up the Jones polynomial for your knot in this knot table and add it to your wiki page. In your justification, simply link to the knot table.

**Read for next class**:

- 6.3

- Another look at the bracket polynomial
- The span of a polynomial
- The span the bracket polynomial for alternating knots

**Homework**:

- 6.9 (you don't have do the entire bracket polynomial, just compute the contribution from 4 different states), 6.10, 6.11
- Pet knot: compute the span of the bracket polynomial for your pet knot. Use the techniques that we saw in the proof of today's main lemma. You do not have to compute the bracket polynomial. You may not use the result of the lemma to answer this question (although it will give you a good check).
- Wiki (due next Monday):
- Fill in the section on the genus of your pet knot
- Look up the Jones polynomial for your knot in this knot table and add it to your wiki page. In your justification, simply link to the knot table.

**Read for next class**:

- 6.3

**Announcements**:

- Remember that a first draft of your paper is due on Thursday.

- The bracket polynomial
- Writhe
- The X polynomial
- The Jones polynomial

**Homework**:

- 6.3, 6.4, 6.5, 6.7, 6.8

**Read for next class**:

- 6.2

- Braid groups
- Markov equivalence
- Markov's theorem
- Knot polynomials
- The bracket polynomial

**Homework**:

- 5.23, 5.27, 5.28, 5.29, 6.1

**Announcement**

- Don't forget that you must answer
**both**questions below in order to receive full class participation points.

**Read for next class**:

- 6.1

- Satellite knots continued
- Cable knots
- Braids
- Closed braids
- Alexander's braid theorem

**Homework**:

- 5.17, 5.18, 5.19, 5.20
- Pet knot: draw a closed braid representation of your knot. Then find the word representing the corresponding braid. (Do this on paper, we'll put it on the wiki later.)

**Read for next class**:

- 5.4

- Prime decomposition theorem for knots
- Torus knots
- Satellite knots
- Whitehead doubles
- Two strand cable knots
- Connected sums as satellite knots

**Homework**:

- 5.1, 5.2, 5.3 (draw a (3,6)-torus link), 5.4, 5.8, 5.13

**Read for next class**:

- 5.2, 5.4

- The genus of a knot
- Crossing number vs. genus
- Different Seifert surfaces for the same knot
- Seifert surfaces for alternating knots
- The genus is additive with respect to connected sums

**Homework**:

- 4.22, 4.24, 4.27
- Pet knot: sketch a Seifert surface for your knot, then compute the genus (do this on paper)

**Read for next class**:

- 5.1

**Announcements**:

- There will be two cars leaving for the conference on Saturday, one at 8:15 and one at 10:15. Passengers should meet in the front of Tome. Don't forget to get a grab-n-go lunch in the morning.

- Genus of a surface
- Surfaces from disks and bands
- Knots as boundaries
- Seifert's algorithm and Seifert surfaces
- The genus of a knot

**Homework**:

- 4.17, 4.19, 4.20
- Your Pet knot: Fill in the unknotting number section on the wiki

**Read for next class**:

- None

- (We skipped compressible surfaces)
- Surfaces with boundary
- Orientability
- The classification of surfaces

**Homework**:

- 4.13, 4.14, 4.15, 4.16
- Please list your project preferences here by the evening of Sunday, March 22

**Read for next class**:

- 4.3

**Announcements**:

- There will be two vehicles going to the EPaDel Conference on March 28. One leaves at 8:15 the other leaves at 12:15. Please let me know which vehicle you will ride in. There will be students giving talks all morning, so I think you'd enjoy being there (if you can get up that early!).

- Surfaces
- Partitions of surfaces
- Euler characteristic
- Euler characteristic for connected sums

**Homework**:

- 4.5, 4.6, 4.7, 4.8, 4.9, 4.10

**Read for next class**:

- 4.2

- Crossing number
- Crossing number and alternating knots
- Surfaces
- Isotopic surfaces
- Homeomorphic surfaces

**Homework**:

- Due Monday: Fix your wiki page
- Due Monday: 3.15, 4.3
- Due Monday: Look up the unknotting number of your pet knot on this web page. Show on paper that your knot has this unknotting number (ie. draw a sequence of pictures showing that after making the crossing switches that it is the unknot).
- Due next Thursday: Put the unknotting number of your pet knot on your wiki page. In the justification box, draw a picture of your knot with the crossings that you need to switch circled. You do not have to draw the sequence of pictures showing that it is the unknot.

**Read for next class**:

- Nothing new to read

**Announcements**:

- here is the web page I mentioned that had references to the unknotting numbers.
- Below you should find my claymation video.
- Have a nice break!

- Bridge number (two definitions)
- 2-bridge knots and rational knots
- Bridge knots and connected sums

**Homework**:

- 3.10. 3.11, 3.12a, 3.14
- Wiki: put the bridge number of your knot on the wiki. In the justification box, draw a picture of your knot showing that this is indeed the bridge number (use Adams' definition of bridge number)
- Wiki: Fix all of the items that I listed on your talk page (this is due the Monday after spring break).

**Read for next class**:

- 4.1

- Unknotting number
- k-equivalent knots

**Homework**:

- 3.1, 3.3, 3.5, 3.6, 3.7, 3.8c,

**Read for next class**:

- 3.3

- Representing knots as signed graphs

**Homework**:

- 2.27, 2.28, 2.29, 2.31 (Type III only)
- Pet knot: create a signed planar graph for your knot and put it on the wiki. In the justification box draw the signed graph on top of your knot projection. You do not have to shade the faces (although you can if you are able to figure it out). Note: there is no place for it on your pet knot page. Put it under the Conway notation.
- Don't turn in: 2.30

**Read for next class**:

- 3.2

- Rational knots/links
- Conway's square dance
- Algebraic tangles
- Pretzel knots
- Graphs and planar graphs

**Homework**:

- 2.18, 2.19, 2.21, 2.22
- Pet knot: the Appendix has the Conway notation for your knot
- On paper
- Draw a projection of your knot from the Conway notation (as Adams did in Figure 2.24)
- If this projection does not look like the standard projection of your knot, then show that it is the same as the standard projection by giving a sequence of pictures

- On the wiki
- Give the Conway notation for your knot and draw the projection that you obtained above (put this picture in the "justification" box)
- You do not have to put the sequence of pictures on the wiki

- On paper

**Read for next class**:

- 3.1

**Announcements**:

- Please let me know by email whether you can go to the EPaDel MAA conference on Saturday, March 26.
- Here are the details:
**The Quaternions — From Sir William Rowan Hamilton to Modern Physics and Topology**, Lou Kauffman, University of Illinois at Chicago - I am not requiring you to go, but I strongly urging you to go. Lou Kauffman is a major figure in Knot Theory. This will be an exciting opportunity. The department will pay for registration and lunch.
- Here is a website with more information about the conference, including an abstract for the talk and a biography of Kauffman: EPaDel
- We can talk about the departure and return times in class on Monday.

- Rational tangles
- Conway's description of rational tangles
- Continued fractions

**Homework**:

- 2.10, 2.12, 2.13, 2.14

**Read for next class**:

- 2.4

**Announcements**:

- TBD

- Watch the rest of the Conway video

**Homework**:

- Read your talk page and make the changes listed there, then complete the colorability and Dowker notation pages on your wiki

- Dowker notation for alternating knots
- Dowker notation for non-alternating knots
- John H. Conway Priestley Award lecture (part 1)

**Homework**:

- For Thursday: 1.10, 2.6, 2.8, 2.9
- For next Monday: Read your talk page and make the changes listed there, then complete the colorability and Dowker notation pages on your wiki

**Read for next class**:

- Nothing new

- Mod p labelings
- Dowker-Thistlethwaite notation for alternating knots

**Homework**:

- 2.3, 2.5
- Compute the Dowker-Thistlethwaite notation for your pet knot

**Read for next class**:

- Section 2.3

- Linking number is an invariant of oriented links
- The absolute value of the linking number is an invariant of links
- Tricolorability for knots and links
- Mod p labeling

**Homework**:

- 1.17, 1.21, 1.23, 1.28(a), 1.29
- Determine if you pet knot is tricolorable
- Not to turn in: 1.25

**Presentations**:

- Volunteers

**Read for next class**:

- 2.1 and 2.2

**Announcements**:

- There is a math/cs chat tomorrow

- Reidemeister moves
- Links
- Oriented links
- Linking number for oriented links

**Homework**:

- Turn in: 1.10, 1.13 (draw a sequence of pictures—not necessarily one Reidemeister move at a time, though), 1.14, 1.15, 1.16
- Update your "pet knot" page as I mentioned in class
- Do not turn in: 1.18/1.19 if you can't come up with these yourself (they're tricky!) look them up online

**Presentations**:

- We need some new volunteers. If you haven't presented yet, please be prepared for Monday

**Read for next class**:

- 1.5

**Announcements**:

- Here are the links to the two papers I mentioned in class
- Here is a link that gives Gauss' integral for the linking number

- Connected sum
- Composite knots
- Prime knots
- Analogies between multiplication of positive the positive integers and the connected sum of knots
- The unknot is not composite
- Prime decomposition theorem for knots
- Oriented knots
- Invertible knots
- The Reidemeister theorem

**Homework**:

- 1.8
- In class I mentioned that we do not actually need both type I Reidemeister moves. By giving a sequence of pictures, show that we can accomplish the Reidemeister move shown below using the other type I move, together with moves II and III.

**Presentations**:

- I'll take volunteers

**Read for next class**:

- Sections 1.4

- The definition of a knot
- Wild knots (see below)
- Alternating knots
- The wiki
- KnotPlot

- 1.6 (the first question only), 1.7 (these starred ones can be tricky!)
- Is your pet knot alternating? Yes, no, or don't know yet? If so, draw an alternating projection.

**Read for next class**:

- Section 1.3

- Knots
- The unknot, trefoil knot, figure eight knot
- (Regular) projections
- Crossing number

**Homework**:

- Don't turn in: 1.3 (try using KnotPlot to show this)
- Turn in: 1.4, 1.5
- Turn in: we didn't give a mathematical definition of a knot (and if you look at the text, the author doesn't either). Try to give a rigorous definition of the term knot (do your best).

**Presentations**:

- I'll take volunteers

**Read for next class**:

- Sections 1.1 and 1.2

**Announcements**:

- Don't forget to answer the two questions below.
- There are some beautiful pictures on the KnotPlot website.