06-1350/Homework Assignment 1

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T53-Negated.jpg

Solve the following problems and submit them in class by October 19, 2006:

  1. Let p be an odd prime. A knot diagram D is called p-colourable if there is a non-constant map ("colouring") from the arcs of D to {\mathbb Z}/p so that at every crossing, the average of the colours of the two "under" arcs is equal to the colour of the "over" arc (calculations in {\mathbb Z}/p, of course).
    1. Show that the newly-defined meaning of 3-colourable coincides with the definition given for this notion in class.
    2. Show that "being p-colourable" is invariant under Reidemeister moves and hence defines a knot invariant.
    3. (Hard and not mandatory) Prove that the (5,3) torus knot T(5,3) (pictured above) is not p-colourable for any p.
  2. Use the recursion formula Failed to parse (unknown function\overcrossing): q^{-1}J(\overcrossing)-qJ(\undercrossing)=(q^{1/2} - q^{-1/2})J(\smoothing)
and the initial condition J(\bigcirc)=1 to compute the Jones polynomial Failed to parse (unknown function\HopfLink): J(\HopfLink)
of the Hopf link and the Jones polynomial Failed to parse (unknown function\righttrefoil): J(\righttrefoil)
of the right handed trefoil knot.
  1. Explain in detail why is the set {knots of genus 3} definable using knotted trivalent graphs.
  2. Explain in detail why is the set {knots of unknotting number 3} definable using knotted trivalent graphs.

Mandatory but unenforced. Find yourself in the class photo and identify yourself as explained in the photo page.

06-1350-ClassPhoto.jpg