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

MIDTERM II

PART II

Problem 1. Fill the table characterizing each of the singularities f(x, y) below

x^{5} − y^{2} x(x^{5} − y^{2}) x^{4} − y^{3}

notation

braid monodromy (project to x)

number of link components page F of the Milnor open book

monodromy of f(x, y) + z^{2} Milnor open book monodromy (in S^{3})

Problem 2.

(1) What can be the topological type of the double covering ramified at three points over a torus with two punctures ?

(2) How many non-isomorphic such ramified covering do exist ?

(3) Is it possible that two such coverings are not isomorphic, but have homeomorphic covering spaces ?

Problem 3. Consider a Heegaard diagram of L(p, q) represented by curves α, β on a torus T.

(1) How to find p ?

(2) Find p if α = a − 2b and β = 3a + b, where a and b are simple closed curves intersecting at one point with the intersection index a ∘ b = 1.

(3) Find q in the above example.

Problem 4. Consider a Lens space L(p, q) obtained by a Dehn surgery defined by a link-chain with three components, with framings 2, 3, 4.

(1) Perform slam-dunk moves and show how to obtain L(p, q) by a rational Dehn surgery along an unknot.

(2) Find p and q.

(3) Find a framed link-chain with negative framings which describes the same L(p, q).

(4) Does L(p, q) admit an orientation reversing homeomorphism ?