Math 309: Introduction to knot theory Assignment 2
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Math 309: Introduction to knot theory
Assignment 2, due February 3 by 11:59 pm.
1. Prove that each of the 6-crossing knots in the knot table has unknotting number 1.
2. Define the “linking number of a knot” as follows: take a knot K, orient it and make a choice of diagram, then assign +1 or − 1 to each crossing, as in the definition of linking number; finally, take the sum of these +1s and − 1s. Show that this procedure does not produce a knot invariant.
3. Recall that McCoy’s theorem states that an alternating knot with unknotting number 1 admits an unknotting crossing in any minimal alternating diagram. Consider the following diagram then:
For this problem, let this represent a knot referred to as K .
(a) Show that, based on the diagram given, the unknotting number of K is at most 2.
(b) Find an alternating diagram for K, and using this give a proof that the unknotting number of K is exactly 2.
4. A simple link is a 2-component link with the property that if it can be isotoped so that there are exactly two crossings between the link components.
(a) Give an example, with proof, of a simple link. And, give an example, with proof, of a 2- component link that is not simple.
(b) Consider a simple link L with the property that each component is a 3-colourable knot. Prove that the link L is 3-colourable.
2023-02-07