METU Department of Mathematics Math 543 Low-dimensional Topology Fall 2013 S.Finashin

Midterm 1 (Part 2, home-task)

due to November 4 2013, 32 points in 6 problems

Name and the student number:

Q1. (4 pts) (a) Show that Figure-8 knot is amphichiral.

(b) Prove or disprove that K#K is chiral, where K is a right trefoil knot.

Q2. (3 pts) Find a braid presentation of the sum of a figure-8 knot with a right trefoil knot. (Give a sketch a write the product of generators σ_{i}.)

Q3. (6 pts) (a) Prove the relation σ_{1}σ_{2}σ_{1}^{−1} = σ_{2}^{−1}σ_{1}σ_{2} using a braid relation in in the Braid group B_{n}, n ≥ 3.

(b) Explain its meaning by sketching the corresponding braid diagrams.

(c) Deduce that the commutator b = [σ_{1}, σ_{2}] represents an unknot.

Q4. (6 pts) (a) Give a braid presentation of the torus knots K_{3,4} and K_{4,3} (write the corresponding product of generators σ_{i}) ? Use Markov moves and Braid relations to show that these knots are isotopic.

(b) Using braid relations and Markov moves show that 3-braids (σ_{1}σ_{2}^{2})^{2} and (σ_{1}σ_{2})^{3} have isotopic closures.

Q5. (4 pts) Find the monodromy matrix for a connected sum of two right trefoil knots using its relation to the Seifert matrix.

Q6. (9 pts) (a)Calculate the Alexander polynomial of a figure-8 knot using three different methods.

(b) Find the determinant and the signature of the figure-8 knot.