SECTION A

1.  (a) State the defining relations of the Alexander-Conway polynomial.

(b) Calculate the Alexander-Conway polynomial of the Hopf link with positive

linking number.

(c) Let L be an oriented link diagram containing a tangle as in Figure 1(a), and let L00  be the link diagram obtained from L by replacing this tangle with the tangle from Figure 1(b), and L1 the link diagram obtained from L by replacing this tangle with the tangle from Figure 1(c).

(a)

(b)

Figure 1.

(c)

Derive a formula relating the Alexander-Conway polynomials of L, L00  and L1 .

(d) Show that the trefoil knot admits a diagram only involving tangles as in Figure 1(a) (possibly with di↵erent orientations), and calculate its Alexander-Conway polynomial.

2.  (a) Apply Seifert’s algorithm to the knot diagram below.

(b) Determine the genus of the resulting surface.

(c) Calculate the genus of this knot by first simplifying it. 

3.  (a) State the Poincar´e-Hopf Theorem.

(b) Assume the following surface has a vector field with exactly one singularity on the interior, and where the vector field is as indicated on the boundary. Determine the index of the singularity.

SECTION B

4. Given a link diagram D, define the double bracket polynomial of D, denoted hhDii,

as the Laurent polynomial in q obtained by the following rules.

− 1

(DP2) hhD1  t D2 ii = hhD1 ii · hhD2 ii, where D1  t D2  is the disjoint union of link diagrams D1  and D2 .

(a) Calculate hhDii for D a diagram of

• the unlink with three components and no crossings.

• the unknot with exactly one positive crossing and no negative crossings.

• the unknot with exactly one negative crossing and no positive crossings.

• the Hopf link with a two crossing diagram. You only need to do one of the two possible diagrams.

• the right-handed trefoil knot with exactly three positive crossings and no negative crossings.

(b) For an oriented link dia(−)gram D denote by n+ (D) the number of positive cross-

Let

5.  (a) State the defining relations of the absolute polynomial QL (x) of a link L.

(b) Determine the absolute polynomial of the unlink Un  with n components for

each n ≥ 2.

(c) Let L1  and L2  be links. Show that

QL1tL2 (x) = QU2 (x) · QL1 (x) · QL2 (x),

where L1 t L2  is the link obtained by disjoint union.

(d) Let K1  and K2  be knots with K2  invertible. Show that QK1+K2(x) = QK1 (x) · QK2 (x),

where K1 + K2  is the composition of the two knots.