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

MIDTERM 1 (MARCH 30)

Problem 1. (20 pts) Consider a gluing scheme for a surface F described by the words abca^{−1}, d^{2}cef, ghfg^{−1}.

(1) determine the number of vertices for such a scheme;

(2) determine the number of boundary components of F;

(3) find the Euler characteristics χ(F) and χ(F̂) (here and below, F̂ denotes a closed surface obtained by filling the holes in F by discs);

(4) classify the surfaces F and F̂ (i.e., recognize them as one of the surfaces F_{g}, F_{g,l}, N_{k}, N_{k,l});

Problem 2. (30 pts) Consider the link L and its spanning surface F shown on the Figure.

(1) Justify if this surface is orientable (presenting your arguments);

(2) determine the Euler characteristic χ(F);

(3) determine the Euler characteristic χ(F̂);

(4) classify F and F̂ as the surfaces of type F_{g}, F_{g,l}, N_{k}, or N_{k,l}.

(5) sketch a set of basic cycles on F;

(6) find the algebraic number of semi-twists for your cycles and determine which of them are disorienting (if there is any).

Problem 3. (20 pts) For the link L in the previous problem

(1) determine the linking numbers between the components of L (oriented like it is shown)

(2) evaluate the Whitney indices of the components of L as they are projected to the plane;

(3) sketch the Seifert circles;

(4) write the code of the Seifert circles.

Problem 4. (10 pts) Sketch a braid b = σ_{2}σ_{4}σ_{1}^{−1}σ_{3}^{−1} with 5 strands and determine the number of components of b̂.

Problem 5. (10 pts) Assume that knots K_{1} and K_{2} can be presented as closed braids with n and m strands respectively. Prove that K_{1}#K_{2} has a presentation as a closed braid with n + m strands.

Problem 6. (10 pts) Assume that a triangulation of a surface F = T^{2}#T^{2}#ℝP^{2} contains 2n triangles. What can be the number of vertices in this triangulation ? (Hint: make use of the Euler characteristic of F.)

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