METU Department of Mathematics Math 534 Low-dimensional Topology Spring 2009

MIDTERM I

Problem 1(7 pts).

(1) Sketch a 2-bridge diagram of the Hopf link and of the (2, 5)-torus knot.

(2) find the twist sequence associated with the number \frac{7}{17} and sketch the corresponding 4-plat.

Problem 2(7 pts). Find a sequence of the Reidemeister moves which shows that (1) the closure of braid σ_{1}σ_{2}σ_{1}^{−1}σ_{2}^{−1} is an unknot;

(2) the 4-plat with the twist sequence (1, 1, −1, 1, 1) is an unlink.

Problem 3(9 pts). Sketch the closure, L, of braid b ∈ B_{4}, b = (σ_{1}σ_{2}σ_{3})^{2}σ_{1}^{2}.

(1) Find the linking number between the component of links L and the writhe of these components.

(2) Sketch the canonical span of L and determine if it is orientable. Find its topological type.

(3) Find the genus of a Seifert surface constructed using the above braid presentation of L.

Problem 4(10 pts).

(1) Find the Alexander polynomial of the torus knot K_{3,4}.

(2) Find its Conway polynomial.

(3) Find its Casson and Arf invariants.

(4) Find its determinant and signature.

Problem 5(12 pts).

(1) Find a braid presentation of knot K_{1}#K_{2}#K_{3}, where K_{1}, K_{2} are righthand and left-hand trefoil knots, and K_{3} is (2, 5)-torus knot.

(2) Simplify braid presentation of knot K = b̂, where b = σ_{1}^{−1}σ_{3}σ_{1}^{2}σ_{2}σ_{3}σ_{2}^{3}σ_{1}^{2}σ_{4}σ_{1}^{3}, using Markov moves, and present K as a connected sum of prime knots.

(3) Give braid presentation of (2, 5)- and (5, 2)-torus knots and show their equivalence using braid relations and Markov moves.

Problem 6(15 pts). Find the Alexander polynomial of the torus knot K_{2,5} using (1) reduced Burau presentation,

(2) Seifert form,

(3) skein relations.