Drorbn at 23:05, 4 October 2006

2006-10-04T23:05:07Z

← Older revision		Revision as of 19:05, 4 October 2006
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	## 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~~%285%2C3%29~~ T(5,3)] (pictured above) is not <math>p</math>-colourable for any <math>p</math>.		## (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.

Drorbn at 23:00, 4 October 2006

2006-10-04T23:00:58Z

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{{06-1350/Navigation}}

[[Image:T53-Negated.jpg|center]]

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

# Let <math>p</math> be an odd prime. A knot diagram <math>D</math> is called <math>p</math>-colourable if there is a non-constant map ("colouring") from the arcs of <math>D</math> to <math>{\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>{\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>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%285%2C3%29 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.
# 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 [[06-1350/Class Photo|photo page]].

[[Image:06-1350-ClassPhoto.jpg|center|400px]]

06-1350/Homework Assignment 1 - Revision history

Drorbn at 23:05, 4 October 2006

Drorbn at 23:00, 4 October 2006