Exercise 3.4.11. (2000F) (i). Let D be a knot diagram. Define the number of p-colourings Tp(D) of D. If D is the trefoil diagram shown below, write down an explicit set of linear equations (over the field Fy of p elements) whose solutions corresponds bijectively to p-colourings of D. (ii). Compute, for each prime number p 3, the number Tp of p-colourings of the trefoil. 37 (iii). Suppose a p-coloured knot diagram D contains a portion like the one shown below, with colours Ii, Yi appearing at the places indicated. What are the equations relating ti+1, Yi+1 to Ii, yi? Let z; = 1; y. What are the equations relating Zi+1, 13+1 to zi:1;? Use this to express In: Zn in terms of 60,20 20 21 22 17-2 -1 OCD Yo y2 Yn-2 Yn-1 Yn (iv). Let m be a positive integer and p > 5 be prime. The (2, 2mp + 3) torus knot can be described by a diagram D of the form shown below, with 2mp +3 crossings. Show that Tp(D) =p. Exercise 3.4.11. (2000F) (i). Let D be a knot diagram. Define the number of p-colourings Tp(D) of D. If D is the trefoil diagram shown below, write down an explicit set of linear equations (over the field Fy of p elements) whose solutions corresponds bijectively to p-colourings of D. (ii). Compute, for each prime number p 3, the number Tp of p-colourings of the trefoil. 37 (iii). Suppose a p-coloured knot diagram D contains a portion like the one shown below, with colours Ii, Yi appearing at the places indicated. What are the equations relating ti+1, Yi+1 to Ii, yi? Let z; = 1; y. What are the equations relating Zi+1, 13+1 to zi:1;? Use this to express In: Zn in terms of 60,20 20 21 22 17-2 -1 OCD Yo y2 Yn-2 Yn-1 Yn (iv). Let m be a positive integer and p > 5 be prime. The (2, 2mp + 3) torus knot can be described by a diagram D of the form shown below, with 2mp +3 crossings. Show that Tp(D) =p