06-1350/Homework Assignment 1: Difference between revisions
From Drorbn
Jump to navigationJump to search
No edit summary |
mNo edit summary |
||
| Line 8: | Line 8: | ||
## Show that the newly-defined meaning of 3-colourable coincides with the definition given for this notion in class. |
## Show that the newly-defined meaning of 3-colourable coincides with the definition given for this notion in class. |
||
## Show that "being <math>p</math>-colourable" is invariant under Reidemeister moves and hence defines a knot invariant. |
## Show that "being <math>p</math>-colourable" is invariant under Reidemeister moves and hence defines a knot invariant. |
||
## (Hard and not mandatory) Prove that the (5,3) torus knot [http://katlas.math.toronto.edu/wiki/T |
## (Hard and not mandatory) Prove that the (5,3) torus knot [http://katlas.math.toronto.edu/wiki/T(5,3) T(5,3)] (pictured above) is not <math>p</math>-colourable for any <math>p</math>. |
||
# Use the recursion formula <math>q^{-1}J(\overcrossing)-qJ(\undercrossing)=(q^{1/2} - q^{-1/2})J(\smoothing)</math> and the initial condition <math>J(\bigcirc)=1</math> to compute the Jones polynomial <math>J(\HopfLink)</math> of the Hopf link and the Jones polynomial <math>J(\righttrefoil)</math> of the right handed trefoil knot. |
# Use the recursion formula <math>q^{-1}J(\overcrossing)-qJ(\undercrossing)=(q^{1/2} - q^{-1/2})J(\smoothing)</math> and the initial condition <math>J(\bigcirc)=1</math> to compute the Jones polynomial <math>J(\HopfLink)</math> of the Hopf link and the Jones polynomial <math>J(\righttrefoil)</math> of the right handed trefoil knot. |
||
# Explain in detail why is the set {knots of genus 3} definable using knotted trivalent graphs. |
# Explain in detail why is the set {knots of genus 3} definable using knotted trivalent graphs. |
||
Latest revision as of 18:05, 4 October 2006
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Solve the following problems and submit them in class by October 19, 2006:
- 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).
- Show that the newly-defined meaning of 3-colourable coincides with the definition given for this notion in class.
- Show that "being [math]\displaystyle{ p }[/math]-colourable" is invariant under Reidemeister moves and hence defines a knot invariant.
- (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].
- 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.
- Explain in detail why is the set {knots of genus 3} definable using knotted trivalent graphs.
- 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.
