Knot theory exercise sheet - week III _1_ Identify the following

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Knot theory exercise sheet - week III _1_ Identify the following Powered By Docstoc
					           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.

				
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