Math 164H Schedule
Fall 2007
Abbi trying to calculate the infamous Alexander polynomial over Thanksgiving break, with help from her dad. Thanks, Mr. Konowitz!
Some useful links: (thanks Jenny) :)
The Knot Atlas
Unknotting numbers
Matrix Calculator
Celtic Knots
Review for final.
Projects due and shared.
Ya'll are fabulous and all deserve a shout-out! Thanks for all of your hard work!
Emalee's project
David's project
Kayla's project
Mark's project
Chet's project
Questions about projects answered.
Instructor evaluations completed
New duedate for the project: Monday, December 3 2007
Alexander polynomial - for Hopf links the orientation matters!
Genus of Seifert Surface
Help with genera: http://www.kurims.kyoto-u.ac.jp/~stoimeno/ptab/index.html (Thanks Chet!)
Homework:
Find the canonical genus of your knot.
New duedate for the project: Monday, December 3 2007
Genus of a surface
Oriented surfaces whose boundary is a knot
Seifert's algorithm
Genus of your knot
Canonical genus of your knot
Homework:
Exercise 4.20 and
Find the canonical genus of your knot.
New duedate for the project: Monday, December 3 2007
Wednesday and Friday, 14 and 16 November
Surfaces
Forming surfaces
Orientable vs. non-orientable
Classification theorem for surfaces
Euler characteristic
New duedate for the project: Monday, December 3 2007
Monday, 12 November
Homework: Find the Alexander polynomial for your knot.
This will be part of your project but will not be collected seperately.
New duedate for the project: Monday, December 3 2007
Friday, 7 November
Alexander polynomial calculations
Homework: Find the Alexander polynomial for your knot.
This will be part of your project but will not be collected seperately.
New duedate for the project: Monday, December 3 2007
Wednesday, 5 November
Bridge number - minimal number of maxima or the minimal number of maximal overpasses.
b(K) = 1 if and only if K is the unknot.
b(K) = 2 if and only if K is a rational knot.
Site for more information:
http://www.popmath.org.uk/exhib/pagesexhib/bridgeno.html
Homework for Friday: Find the bridge number for your knot.
New duedate for the project: Monday, December 3 2007
Monday, 5 November
The Crossing Matrix and its Determinant
Terrell's test for m-colorability
Use http://www.bluebit.gr/matrix-calculator/ to calculate the determinant.
Homework for Wednesday: For which primes is your knot p-colorable? Use Terrell's test to figure this out.
New duedate for the project: Monday, December 3 2007
Friday, 2 November
Torus Knots, Hyperbolic Knots, and Satellite Knots
Hyperbolic space
Hyperbolic knots
Homework for Friday: Is your knot a torus knot? Explain.
New duedate for the project: Monday, December 3 2007
Torus Knots, Hyperbolic Knots, and Satellite Knots
Hyperbolic space
Escher
Dancing Building:
Picture 1
Picture 2
Homework for Friday: Is your knot a torus knot? Explain.
New duedate for the project: Monday, December 3 2007
Mathematically celtic knots and knot plot.
New duedate for the project: Monday, December 3 2007
Mathematically celtic knots and knot plot.
Homework for Monday - Is your knot mathematically celtic? Show me how!
New duedate for the project: Monday, December 3 2007
Conway notation. Finding Conway notation for your knots.
Mathematically celtic knots.
Homework for Friday - If you haven't found it, discover the Conway notation (if there is any) for your knot.
Homework for Monday - Is your knot mathematically celtic? Show me how!
New duedate for the project: Monday, December 3 2007
See Bob Dole presentation. Make sure that you get a form for being there and turn it in at the end!
Homework - Discover the Conway notation (if there is any) for your knot - due Wednesday.
Conway Notation
Conjectures formed: An odd number of odd numbers added together yields a knot.
A single even with any number of odds forms a knot.
Negative numbers do not affect the formation of knots vs. links.
Good work!
Homework - Discover the Conway notation (if there is any) for your knot - due Wednesday.
Conway notation
Multiplication of tangles
Addition of tangles
Homework - Which additions (in Conway notation) give a knot (as opposed to a link)?
Conway notation
Continued Fractions
Homework - Exercise 2.11 - Use R-moves to show that the 2 1 1 tangle is the same as the -1 -2 2 tangles are equivalent.
Exam 1
Review sheet for Exam 1
Review for Exam 1
Review sheet for Exam 1
Dowker notation that creates composite knots.
Dowker notation that indicates a Type II R-move.
Drawing knots from given Dowker notations.
Dowker notation indicating Type I R-moves.
Homework:
Investigate when Dowker notation indicates a Type II R-move.
Dowker notation - starting at an undercrossing, finding Dowker notation for different knots.
Dowker notation for non-alternating knots.
Recreating knots from Dowker notation.
Homework:
Find all possible Dowker notations for your knot.
Groups to help with finding a p-coloring for your knot (for some prime p).
Homework:
Find a prime p (other than 3) so that your knot is p-colorable.
Finding a p-coloring for your knot (for some prime p).
Homework:
Find a prime p (other than 3) so that your knot is p-colorable.
Discuss Exercises 1.23 and 1.25.
Modular arithmetic.
p-colorings.
Homework:
Find a prime p (other than 3) so that your knot is p-colorable.
Tricolorability is a knot invariant.
That is, it is invariant under Reidemeister moves.
We showed that this is true for Type I and Type II moves.
Homework:
Show that tricolorability is invariant under Type III moves. (Exercise 1.23)
Show that tricolorability is preserved under composition. (Exercise 1.25)
Tricolorability
Homework:
Give a sequence of moves transforming the given knot into the unknot. Be careful and show all steps.
Is your knot tricolorable?
Reidemeister Moves
Type I, Type II, and Type III moves
Planar Isotopy
Reidemeister's theorem
Homework:
Give a sequence of moves transforming the given knot into the unknot. Be careful and show all steps.
Monday, 17 September
Discuss the three articles distributed:
Biography of Tait
History of Knot Theory
First 1.7 Million Knots
Friday, 13 September
Read the three articles distributed:
Biography of Tait
History of Knot Theory
First 1.7 Million Knots
Wednesday, 12 September
Unknotting numbers of composite knots.
Which torus knots are knots and not multiple component links?
Homework: Read the three articles distributed:
Biography of Tait
History of Knot Theory
First 1.7 Million Knots
Monday, 10 September
Unknotting numbers of composite knots.
Definition of torus knots.
Definition of links.
Homework: Try to answer the following three questions:
(Sandi) Is u(K#L) = u(K)+u(L)?
(Jenny) Is u(K#L) possibly greater than u(K)+u(L)?
(Abbi) Does it matter if both factor knots are alternating?
For which p and q are T(p,q) knots (instead of multiple component links)?
Friday, 7 September
Composite knots.
Alternating projections are composite if and only if they are obviously composite.
Homework: Try to answer the following three questions:
(Sandi) Is u(K#L) = u(K)+u(L)?
(Jenny) Is u(K#L) possibly greater than u(K)+u(L)?
(Abbi) Does it matter if both factor knots are alternating?
Wednesday, 5 September
Pictures of Poland.
Monday, 3 September
Labor Day - No Classes
Friday, 31 August
Unknotting number is an invariant.
Definition of invariant.
Discussion of unknown unknotting numbers.
Wednesday, 29 August
Discussion of algorithm for unknotting.
Discussion of unknotting numbers
Homework: Find the unknotting number of your knot.
Write up a careful argument for this.
This will be collected on Friday.
Monday, 27 August
Discussion of how to unknot your knots.
We asked three questions.
Q: Can every knot be unknotted by changing crossings? YES.
Q: Is there an algorithm for unknotting a know? YES.
Q: What is the minimal number of crossings that need to be changed to unknot your knot? (i.e., what is the unknotting number of your knot?)
Homework: Find the unknotting number of your knot.
This will be collected on Wednesday.
Friday, 24 August
Discussion of Homework questions (good job Chet and Mark!)
How to change knots into unknots by changing crossings.
Homework: Convince yourself (and me) that you can change some number of crossings in any knot to make it into the unknot. (Hint - do this first for your knot, and then try to come up with a general algorithm of how to do this for any knot.)
I will collect this on Monday.
