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

HOMEWORK II (DUE TO MARCH 15)

KNOTS AND LINKS

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. Sketch a torus knot K_{4,1}^{+} and link K_{4,2}^{+}.

(1) Expose a deformation showing that K_{4,1}^{+} is an unknot.

(2) Determine the linking number between the two components of K_{4,2}^{+}.

(3) Show that the components of K_{4,2}^{+} can be permuted by a deformation.

(4) Show that there is no deformation of K_{4,2}^{+} to itself which preserves the orientation of one component and reverses the orientation of the other one.

Problem 2. Sketch a knot diagram for K = K_{3,4}^{+}

(1) Sketch the mirror image \overline{K} and the inverse −K and determine which code K_{m,n}^{±} describes them.

(2) Is K a 3-colorable knot ?

(3) Sketch an unknot with the same projection as K.

(4) Show that U(K) ⩽ 4, that is specify 4 crossings which should be modified to get a knot diagram of a unknot.

Problem 3. Sketch a diagram of K = K_{2,2}^{+}(K_{2,3}^{−}), that is a right (2, 2)-cable link, whose core is a left trefoil. Determine the linking number between its components.

Problem 4. Sketch the positive Whitehead double K = W^{+}(K_{2,3}^{+}#K_{2,3}^{−}) marking the positive crossings on its diagram.

Problem 5. Prove that for any knots K_{1} and K_{2} (1) c(K_{1}#K_{2}) ⩽ c(K_{1}) + c(K_{2}),

(2) u(K_{1}#K_{2}) ⩽ u(K_{1}) + u(K_{2}).

Typeset by AMS-TEX