06-1350/Homework Assignment 1

06-1350/Navigation Panel   # Week of... Links 1 Sep 11 About, Tue, Thu 2 Sep 18 Tue, Kurlin(P), Thu 3 Sep 25 Tue, Photo, Thu 4 Oct 2 HW1, Tue, Thu 5 Oct 9 Tue(P), Thu(P) 6 Oct 16 HW2, Tue(P), Thu(P) 7 Oct 23 Tue(P), Thu 8 Oct 30 HW3, Tue, Thu 9 Nov 6 Tue ([math]\displaystyle{ \nu }[/math]), Thu 10 Nov 13 Tue, Thu 11 Nov 20 HW4(P), Thu 12 Nov 27 Thu 13 Dec 4 Syzygies in Asymptote, Final Jan 8 Grades Note. (P) means "contains a problem that Dror cares about". Add your name / see who's in! On to 07-1352

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

1. Let [math]\displaystyle{ p }[/math] be an odd prime. A knot diagram [math]\displaystyle{ D }[/math] is called [math]\displaystyle{ p }[/math]-colourable if there is a non-constant map ("colouring") from the arcs of [math]\displaystyle{ D }[/math] to [math]\displaystyle{ {\mathbb Z}/p }[/math] 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 [math]\displaystyle{ {\mathbb Z}/p }[/math], 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 [math]\displaystyle{ p }[/math]-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 [math]\displaystyle{ p }[/math]-colourable for any [math]\displaystyle{ p }[/math].
2. Use the recursion formula [math]\displaystyle{ q^{-1}J(\overcrossing)-qJ(\undercrossing)=(q^{1/2} - q^{-1/2})J(\smoothing) }[/math] and the initial condition [math]\displaystyle{ J(\bigcirc)=1 }[/math] to compute the Jones polynomial [math]\displaystyle{ J(\HopfLink) }[/math] of the Hopf link and the Jones polynomial [math]\displaystyle{ J(\righttrefoil) }[/math] 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.