Middle East Technical University Low-dimensional Topology Math 710 Spring 2005

Midterm I

(take-home)

Knots, Links and Braids

Problem 1. Prove that the Alexander polynomial of a knot is symmetric.

Problem 2. For the figure-8 knot K (1) justify that K is reversible, obvertible and invertible;

(2) sketch a Seifert surface for K, determine its genus and the associated Seifert matrix, S;

(3) find Δ_{K} and σ_{K} using S;

(4) determine the vertex/region matrix of K and find Δ_{K} using it;

(5) find Δ_{K} using skein relations;

(6) find the Casson invariant and the Arf invariant of K;

(7) sketch the chord diagram of K;

(8) determine the fundamental group of K;

(9) present K as b̂ for some braid b and write down b as a product of generators σ_{i} and their inverse;

(10) evaluate the Kauffman bracket [K] and the polynomyal f_{K}(A).

Problem 3. Sketch the satellite knots, K_{1}, K_{2}, for the left trefoil K̄_{2,3} using as the patterns trefoil knots K_{2,3} and K_{3,2}. Write down the word presentation of the braids whose closures are K_{1} and K_{2}.