Knot theory exercise sheet - week III (1) Identify the following knots. That is ﬁnd knots in the table of knots which the following ones are equivalent to. (2) Look at the table of knots. Each knot is drawn as if it con- sisted of two “parallel” thin strings. Think of this as of the diagram of a link with two components. Calculate the linking number for all such links from the ﬁrst page of the table. Try to work out a general method of determining the linking number for such links. (3) Are the following links splitable? (4) Determine which sentences are true or false (prove the true ones and ﬁnd a counterexample to each false one): (a) A two component link has zero linking number if and only if it is splitable. (b) The knot (2n+1) 1 with one crossing changed is equivalent to the knot (2n − 1) 1, where n > 1. (c) There exists a knot which can’t be coloured mod n for any n ∈ N. (d) For any n > 1 there exists a knot which can be coloured mod n. (5) A link is called alternating if it has a diagram with if the cross- ings alternate under, over, under, over, as you travel along each component of the link. Identify all non-alternating knots in the table of knots. (6) Calculate the determinant of all knots up to seven crossings. (7) Calculate the determinant of all non-alternating knots with up to eight crossings. (8) Calculate the determinant of the Hopf link, the Whitehead link and the Borromean rings. (9) Find a knot with the determinant equal to zero. What can you say about the colouring properties of this knot? (10) Go to the St Machar Cathedral cemetery and admire the links on graves. Think about the linking numbers, splitability, coloura- bility and the misery of human’s life.