Knot theory exercise sheet - week V (1) Determine the colouring ...

Knot theory exercise sheet - week V
(1) Determine the colouring group for any 4 non-trivial knots from
the table.
(2) Determine the colouring group for the Hopf link, Whitehead
link and the Borromean rings. What is the determinant of
these links?
(3) Determine the colouring group of the following knot
Can you identify this knot in the table of knots? What is
the determinant of this knot?
(4) Give an example of a knot or a link with the colouring group
isomorphic to:
(a) Z
(b) Z 2
(c) Z n
(d) Z/3
(e) Z/5
(f) Z/7
(g) Z/p, where p is a prime number.
(h) Z/3 ⊕ Z/5
(i) Z/15
(j) Z/2
(5) Calculate the Alexander polynomial for the figure eight knot, a
knot with 7, 8 and 9 crossings and for the knots 3 1, 5 1, 7 1
and 9 1.
(6) Determine the Alexander polynomial for the Borromean rings.
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(7) Is the following link splitable?
(8) Express the following knot as the sum of prime knots and determine
its Alexander polynomial.
(9) Suppose that the polynomial t 4 +5t 3 +5t 2 +5t+5 is the Alexander
polynomial of a knot K. Is K prime?