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 ($\nu$), 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 $p$ 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"): {\displaystyle 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 (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.