in discussion Daily postings / Homework and discussions » Thursday, April 30

Presentations today. No homework.

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divbyzero 01 May 2009 20:36

in discussion Daily postings / Homework and discussions » Thursday, April 30

in discussion Daily postings / Homework and discussions » Thursday, April 30

Presentations today. No homework.

divbyzero 27 Apr 2009 20:52

in discussion Daily postings / Homework and discussions » Monday, April 27

in discussion Daily postings / Homework and discussions » Monday, April 27

**Topics from today's class**:

- 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.

divbyzero 24 Apr 2009 20:35

in discussion Daily postings / Homework and discussions » Thursday, April 23

in discussion Daily postings / Homework and discussions » Thursday, April 23

**Topics from today's class**:

- 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

it seems we don't have a neat way to easily prove that a arbitrary knot IS amphicheiral.

Re: What did you find confusing or what would you like to know more about? by Tans, 23 Apr 2009 14:43

medinae 23 Apr 2009 04:54

in discussion Daily postings / Homework and discussions » Monday, April 20

in discussion Daily postings / Homework and discussions » Monday, April 20

Is it possible for a knot to have a palindromic X polynomial but not be amphicheiral?

What is the "signature" invariant that Adams refers to on p. 178?

Re: What did you find confusing or what would you like to know more about? by medinae, 23 Apr 2009 04:54

medinae 23 Apr 2009 04:52

in discussion Daily postings / Homework and discussions » Monday, April 20

in discussion Daily postings / Homework and discussions » Monday, April 20

There are clever ways for determining whether or not a knot is amphicheiral; for example, the X polynomial of an amphicheiral knot must be palindromic. An interesting fact about amphicheiral alternating knots is that they must have an even crossing number, but knots in general may have an odd crossing number and be amphicheiral.

Other than by using polynomials, what are the other methods for determing if a knot is ampherical?

Re: What did you find confusing or what would you like to know more about? by deweyo, 23 Apr 2009 03:34

How was the concept of amphicerality even worked with before polynomials?

Re: What did you find confusing or what would you like to know more about? by rodya, 23 Apr 2009 01:08

Mike Landis 22 Apr 2009 21:02

in discussion Daily postings / Homework and discussions » Monday, April 20

in discussion Daily postings / Homework and discussions » Monday, April 20

The polynomial of an amphicheiral knot is palindromic, which implies that zeros must be listed along with all other coefficients.

Mike Landis 22 Apr 2009 20:59

in discussion Daily postings / Homework and discussions » Monday, April 20

in discussion Daily postings / Homework and discussions » Monday, April 20

If Max Dehn didn't have polynomials to work with, what did his proof of the distinction between right and left trefoil knots consist of?

Re: What did you find confusing or what would you like to know more about? by Mike Landis, 22 Apr 2009 20:59

divbyzero 20 Apr 2009 21:33

in discussion Daily postings / Homework and discussions » Monday, April 20

in discussion Daily postings / Homework and discussions » Monday, April 20

Note: this forum is not a substitute for my office hours—if you are having trouble, please come visit me.

What did you find confusing or what would you like to know more about? by divbyzero, 20 Apr 2009 21:33

in discussion Daily postings / Homework and discussions » Monday, April 20

Write one sentence about the reading assignment. You cannot repeat what another student has posted.

in discussion Daily postings / Homework and discussions » Monday, April 20

**Topics from today's class**:

- 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)

what's the goodness of using two invariants instead of one?

Re: What did you find confusing or what would you like to know more about? by Tans, 20 Apr 2009 14:47

The HOMFLY polynomial is not a complete invariant, for example, it can not distinguash mutant knots.

Mike Landis 20 Apr 2009 12:56

in discussion Daily postings / Homework and discussions » Thursday, April 16

in discussion Daily postings / Homework and discussions » Thursday, April 16

The explanation of the relationship between the braid index of a knot and its HOMFLY polynomial was somewhat confusing.

Re: What did you find confusing or what would you like to know more about? by Mike Landis, 20 Apr 2009 12:56

Mike Landis 20 Apr 2009 12:55

in discussion Daily postings / Homework and discussions » Thursday, April 16

in discussion Daily postings / Homework and discussions » Thursday, April 16

The HOMFLY polynomial was the first polynomial to rely on two variables.

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