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

HOMEWORK III (DUE TO APRIL 9)

BRAIDS AND SEIFERT SURFACES

Please, give your solution to 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. Find a braid b whose closure, b̂, presents (1) Borromean rings,

(2) the twisted Whitehead double W_{2}^{+} Write b as a product of the generators σ_{i} and σ_{i}^{−1}.

Problem 2.

(1) Sketch an oriented span for the Whitehead link, L. (Hint: You may find a suitable diagram for the Whitehead link in the solution to Problem 2.4 in Prasolov’s book.)

(2) Find the corresponding Seifert matrix.

(3) Find the genus of this Seifert surface.

(4) Find the signature, σ(L).

Problem 3. For the knots K_{1} = K_{2,3}^{+}#K_{2,3}^{+} and K_{2} = K_{2,3}^{+}#K_{2,3}^{−}, determine (1) the Arf invariants,

(2) the Alexander polynomials,

(3) the signature.

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