Lab 8

Exercise 1

• The reflection of a knot is the one we get when reflecting on a plane not intersecting the knot.

• If K is an oriented knot, the reverse of K is the same knot with the opposite orientation.

• Show that the knot 4_1 (the figure-eight knot) is equivalent to its reverse and reflection.

Exercise 2

Compute the linking number of the Hopf link with all possible orientations.

Exercise 3

Show that the Jones polynomial of an oriented link L takes the value (-2)^{#L-1} when a=1, where #L is the number of components of L.

Exercise 4

• Compute the writhe of the knot 4_1 by hand.
• Compute the Kauffman bracket of the knot 4_1 by hand.
• Deduce the Jones polynomial of the knot 4_1.

Exercise 5

• In Python or Sage/CoCalc, load the knot 4_1 and compute its Jones and HOMFLY polynomials. Compare it to the one you computed in Exercise 4.
  Hint: you may enter it in CoCalc as K1 = Link([[8,3,1,4],[2,6,3,5],[6,2,7,1],[4,7,5,8]])
• Verify that the Jones polynomial is a specialisation of the HOMFLY polynomial.
• Help: https://doc.sagemath.org/html/en/reference/knots/sage/knots/knot.html