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 be an odd prime. A knot diagram is called -colourable if there is a non-constant map ("colouring") from the arcs of to 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 , 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 -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 -colourable for any .
  2. Use the recursion formula Failed to parse (unknown function "\overcrossing"): {\displaystyle q^{-1}J(\overcrossing)-qJ(\undercrossing)=(q^{1/2} - q^{-1/2})J(\smoothing)} and the initial condition to compute the Jones polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J(\HopfLink)} of the Hopf link and the Jones polynomial Failed to parse (unknown function "\righttrefoil"): {\displaystyle J(\righttrefoil)} of the right handed trefoil knot.
  3. Explain in detail why is the set {knots of genus 3} definable using knotted trivalent graphs.
  4. 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