The Jones Polynomial: Section 4.4

1. The Jones Polynomial:Section 4.4 • Chris Thiel • Math 552

2. As seen on past episodes...

3. Or if you tilt your head 90 degrees: The unknot: The Unlink: The Kauffman Bracket

4. For example

5. Keep your shirt on, Slick! Have you already forgotten about writhes? Great... if I want to avoid Reidemeister I moves the rest of my life!

6. The Writhe For the sake of invariance in R1, we orient our diagram, and add up the positive and negative crossings

7. The Hopf Link now we normalize with the writhe, if the links are oriented the same direction the writhe is 2, if you orient the links in the opposite directions, the writhe is -2

9. Consider the Left Trefoil now we substitute Any direction we choose to orient this diagram get a writhe w(D)=-3 and compute

10. Consider the Right Trefoil now substitute Any direction we choose to orient this diagram get a writhe w(D)=+3 and compute

11. Jen says you can find the Jones Polynomial with AXIOMS instead of the BRACKET! That’s SO dreamy!

12. That’s right, and you can too, if you know the Jones skein relation: And of course, the Jones Polynomial of the Unknot is 1! (see Th 4.4.3 page 26)

13. Unlink Say, aren’t those first two unknots?

14. Hopf Link

15. Say, that IS nifty! I guess it works by some sort of MAGIC! I suppose you think all this ironing is done by magic, too.

16. Now we normalize each bracket with the writhe so we replace each Let see what’s going on by looking at the diagrams of the skein relation:

17. Careful! Pick a crossing where BOTH arrows can go up! Left Trefoil

18. Left Trefoil

19. Right Trefoil

20. Looks like the other one, with all the signs in the exponents switched Wake up, Bob! That was the second time we did that Polynomial!

21. Let’s spell it out: Mirror Image? Swap the signs of the exponents! Proof: The mirror image negates the writhe of any oriented diagram (Exchanging the positive and negative crossings). The effect on the Kauffman Bracket is that A is replaced with 1/A.

22. You’re speaking in palindromes again, dear! Was it a car or a cat I saw? Hell its a Toyota still, eh? That can’t be right...What about the mirror image of the Pentafoil? Its Jones polynomial is DammitImmad!

23. Figure 8 (Bracket Method)

24. Next substitute Figure 8 (Bracket Method) Now compute the writhe and normalize

25. Of course, Dear! The Jones skein method is like cooking in the “Kitchen of Tomorrow” Mommy, that took FOREVER! Isn’t there a quicker way?

26. Note the orientation! The Negative Hopf! Figure 8-skien method

27. There’s a cute trick for the Mirror Image:a cyclic permutation switches crossings! We many run out of letters so the standard Planar Diagram(PD) uses numbers,so here the left trefoil is Computer Methods- See Kauffman’s “Bracket Calculations.pdf” pp 5-8 http://www.math.uic.edu/~kauffman/MLinks.pdf For each crossing, start with a end on an overcrossing and go counterclockwise Other Encodings include the Gauss Code and the “Dowker-Thistlewaite (DT)” Code http://www.math.toronto.edu/~drorbn/KAtlas/Manual/index.html

28. If only I could figure how to put a knot on a punch card!

29. Connected Sums K=[daeb][becf][gdfc][gkhj][kilh][jlia] According to my Mathematica Calculations:

30. Bao’s knot from last time: Rolfsen’s 6.1 BaosKnot = [ibha] [chbg][dlcm][ldke][ekfj][fmgn][niaj] K61 = [kbja] [bicj][dgeh][flgk][idhc][kbja] Does Same Jones Polynomial Mean it’s the Same Knot?

31. Well, since it’s under 10 crossings, yes. According to Alexei Sossinsky, these non-isotopic prime knots have the same polynomial

32. •Odd number of components: •Even number of components: Observations so far • The Jones Polynomial can spot a amphichiral knot • It can distinguish between the Right and Left Trefoil • It can be computed without the Bracket • The Jones polynomial of the connect-sum is the product of the parts’ Polynomials • There are different Knots with the same Jones Polynomial, but the Unknot (so far) is unique

33. Well, thank heavens! Now we can keep up with the Jones’ Just you wait, Madge, There’s an even NEWER Polynomial out there!

34. HOMFLY HOMFLY J. Hoste, A. Ocneanu, K. Millett, P. Freyd, W. B. R. Lickorish and D. Yetter, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985) 239-246. Unlink Note: There are other names or signs used for these variables

35. HOMFLY: Hopf+

36. Compare this with HOMFLY: Hopf-

37. HOMFLY:Left Trefoil

38. Compare this with HOMFLY: Right Trefoil

39. Hey, I’m no dummy! The Jones was based on the Bracket, but the HOMFLY is based on the axioms directly... it looks good on paper, but how will it play before mathematicians?

40. Number of crossings Minimal number of crossing changes to make the diagram an unlink Complexity of a link diagram D:(c,m)

41. a b a b c d c d For any link, there is a diagram D with a chosen crossing C such that L is either • certainly L0 has lower complexity • a change of crossing will keep the number of crossings the same • a change of crossing might raise the unlinking number, if so let L be the higher of the two associated to D and C. Claim:the other two will have lower complexity

42. I guess to show you that the right choice of a crossing will guarentee the skein relation will give you results. Good thing too, if we expect to blast off at 6:00! What’s the point of that check list?

43. Suppose I is a -valued function of oriented links that is: 1. an invariant of oriented links lying in 2. satisfies the HOMFLY skein relation 3. I(unknot)=1 then I is the HOMFLY polynomial.(that is, the HOMFLY polynomial is characterized uniquely by these three properties

44. smaller complexity so... By induction on the minimal complexity of a link L •True for the unknot. Suppose D had complexity (c,d) andsuppose if complexity (D’)<complexity (D)then I(D’)=P(D’). Show I(D)=P(D) Case if L+ is highest complexity... Take D=L+ Similarly true in the case of L- has highest complexity... So the HOMFLY is uniquely defined by the three properties

45. Gee whiz, that’s swell, But what would I do with yet ANOTHER polynomial in my life? With a HOMFLY you can: *determine the Jones Polynomialof a link *find the Conway Potential Function of a link *compute the Alexander Polynomialof a link *win friends and influence people Polynomials for Fun & Profit

46. to find the Jones Polynomial let to find the Conway Potential Function, let to find the Alexander Polynomial, let

47. Notice there is no constant term(It’s not a knot) Notice the coefficient of zis the linking number (It’s not a knot) (just let x=1 and z= ) Now you try: If the HOMFLY of the Hopf+ Link isfind the Conway Potential Function (just let x=1) Now find the Alexander Polynomial of the Hopf+ Link Notice the Alexander polynomial will never have negative powers of z

48. (a) (b) (c) (d) Homework Question 1: Which is Vaughan Jones, the inventor of the Jones Polynomial?

49. Homework Question 2: Compute the HOMFLY of the Figure 8 Knot, then • Use this to find the Jones (check with the result we found using other methods) • Find the Conway (Should there be a constant term?) • Find the Alexander Polynomial (Can the Alexander Polynomial distinguish between the left and right trefoil?)