Middle East Technical University Spring 2002 422 Elementary Geometric Topology Instructor: S. Finashin

HOMEWORK I (DUE TO MARCH 1)

Please, give your solution for each problem on a separate page. Do not forget to write your name and student’s number on every page that you submit

Problem 1. Prove that the curves with the codes ⟨1⟨1⟨1⟩⟩⟩ and ⟨1⟨2⟩⟩ are not ambient homeomorphic in the plane, deducing it from Jordan’s theorem.

Problem 2. Determine the rooted trees having the following codes (1) ⟨5⊥⊥1⟨2⟩⊥⊥1⟨2⟩⟩

(2) ⟨3⊥⊥1⟨1⟨1⟨2⟩⊥⊥1⟨3⟩⟩⟩⟩

(3) ⟨1⟨2⊥⊥1⟨2⊥⊥1⟨1⟩⟩⟩⟩ Sketch multi-component curves on a plane with such schemes and estimate their minimal possible degree applying Bezout theorem to suitable lines.

Problem 3. For a generic immersed curve γ sketched below (1) find the rotation number;

(2) sketch the multi-component curve obtained by smoothing the double points of γ in accord with the orientations; find the rooted forest and the code of the obtained multi-component curve;

(3) point out the winding numbers for the regions into which γ divides a plane;

(4) sketch a chord diagram of γ with oriented chords and decorate the arcs of circle with the indices.

(5) check your calculation of the rotation number using the decorated chord diagram.

(6) Sketch a deformation of γ to a curve in a standard form. Specify the types of Arnold’s moves involved into your deformation.

Typeset by AMS-TEX