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Presentations today. No homework.

Thursday, April 30 by divbyzerodivbyzero, 01 May 2009 20:36

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.

Monday, April 27 by divbyzerodivbyzero, 27 Apr 2009 20:52

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
Thursday, April 23 by divbyzerodivbyzero, 24 Apr 2009 20:35

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

Amphicheiral knot is in fact the same as the other projection. Their properties such as bracket polynomial are the same also.

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?

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?

The HOMFLY polynomial is helpful determining whether or not a knot is ampherical but it cannot be used in all cases.

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

There are knots which have the same HOMFLY polynomial as their mirror image but which are not amphiceral.

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

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?

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

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

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)
Monday, April 20 by divbyzerodivbyzero, 20 Apr 2009 21:33

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

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

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

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

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