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

Midterm 1

November 6 2013, 17:40 – 19:00, 44 points in 8 problems

Name and the student number:

Q1. (10 pts) (a) Find a braid presentation of the Borromean rings.

(b) Find the Euler characteristics of the natural spanning surface F for a link diagram Borromean rings. Is F orientable ? What is the closed surface F̂ obtained by filling the holes in F with discs ?

(c) Sketch the Seifert circles for the Borromean rings. What is the genus of the corresponding Seifert surface ?

Q2. (6 pts) (a) Give an example of a knot of genus 100, and of a knot with the unknotting number 100.

(b) Sketch a Whitehead double of a figure-8 knot. Find the writhe of your diagram.

Q3. (5 pts) (a) Find the number of components of the link b̂ represented by b = (σ_{1}σ_{2})^{3}σ_{1}^{2}.

(b) Find the linking numbers between each pair of the components.

Q4. (3 pts) Explain why a connected sum of two trefoil knots is not isotopic to a connected sum of two figure-8 knots ?

Q5. (7 pts) (a) Explain why polynomials t^{3} − t + 1 and 2t^{2} + t + 2 cannot be Alexander polynomials of some knots.

(b) Explain why a (3 × 3)-matrix cannot be a Seifert matrix of a knot.

(c) Why ( \begin{matrix}1 & 2 \\ 3 & 4 \end{matrix} ) cannot be a Seifert matrix of a knot ?

Q6. (3 pts) What is the relation between the Seifert form S(x, y) and the monodromy f(x) of an algebraic link ? (Give a formula, do not prove it.)

Q7. (5 pts) Find a Seifert surface and determine the corresponding Seifert matrix of a torus knot K_{2,5}.

Q8. (5 pts) Give a braid presentation of the torus knots K_{2,5} and K_{5,2}. Use Markov moves and Braid relations to show that these knots are isotopic.