Exercise Set VIII MATHS 441
Due: November 11, 2009 Dr. Fischer
1. Consider the knot 52which is depicted on page 126 of your textbook.
(a) Draw a complete diagram of all states for this knot.
(b) Use your diagram to compute the Kauffman polynomial F52(t).
2. (a) Find a resolving tree for the knot 52.
(b) Find a resolving tree for the knot 63.
(c) Use your resolving tree to find F52(t) and P52(x, y).
3. Let ¯
Kdenote the mirror image of the knot K. Let FK(t) and PK(x, y) denote
the Kauffman polynomial and the HOMFLY polynomial of K, respectively.
(a) Use the description in terms of states to show that F¯
K(t) = FK(t−1).
(b) Use skein relations to prove that P¯
K(x, y) = PK(x−1, y).
(c) Is the knot 52amphicheiral?
4. Prove that all our knot polynomials are invariants of unoriented knots.
5. This problem indicates the various connections between our polynomials.
(a) Show that FL(t) can be obtained from PL(x, y) by substituting x=t4√−1
and y= (t2−t−2)√−1.
(b) Show that ∇L(z) can be obtained from PL(x, y) by substituting x=√−1
and y=z√−1.
(c) Prove that VL(s) = FL(s−1/4) and that ∆L(w) = ∇L(w1/2−w−1/2).
6. Suppose that the link Lcan be represented by a diagram L1∪L2with links L1
and L2that are on different sides of some vertical line in the plane. Prove that
the Conway polynomial ∇L(z) = 0.
[See the reverse side for hints!]
