Math 164H Schedule
Fall 2008
Knot
Atlas
Wednesday, 3
December
Alexander polynomials
Practice Final Exam
Monday,
1 December
No class
Monday,
24 November
Projects
due!!! Skip
Alexander polynomial, genus, and torus knot portions of the assignment,
but add in the chirality discussion.
All projects are due
in class. No late projects will be accepted.
Friday,
19 November
Alexander polynomials
Wednesday,
17 November
Celtic knots.
Evaluations of teaching.
Monday,
17 November
Celtic knots.
Celtic knot
slides.
Homework:
Determine if your knot is Celtic or not and show me how.
Monday,
17 November
Celtic knots.
Friday,
14 November
More about Conway notation.
Wednesday,
12 November
Work on your projects.
Monday,
10 November
More about tangles - drawing complicated tangles.
Homework:
Find the Conway notation for your knot.
Which Conway notations give knots? Which give links?
Friday,
7 November
Tangles - drawing complicated tangles.
Homework:
Complete exercise 2.11, using Reidemeister moves to show that the 2 1 1
and -1 -2 2 tangles are the same.
Friday,
7 November
Knots with bridge
number 2.
Amphichiral
Knots
Homework:
Use Reidemeister moves to show that the figure-8 knot is amphichiral.
Wednesday,
5 November
More about unknotting
number and crossing number for composite knots.
Bridge number.
Homework:
Find the Bridge Number for your knot.
Monday,
3 November
If a knot is
alternating, then it's
crossing number is equal to the number of crossings in a reduced
alternating projection of that knot. Any two alternating
knots
can be composed into an alternating knot, so the crossing number of the
composite is the sum of the crossing numbers when both factor knots are
alternating.
Continue to investigate:
What can you say about the unknotting number of a composite knot?
What can you say about the crossing number of a composite knot?
Friday,
31 October
Continuation of
Blake's conjecture - we might have found a counter-example.
Sigh.....
Discussion of how you can recognize composite knots in DN.
Continue to investigate Blake's conjecture.
What can you say about the unknotting number of a composite knot?
What can you say about the crossing number of a composite knot?
Wednesday,
29 October
Discussion of Blake's
conjecture. New questions:
Is it true that every DN that satisfies the pairing condition yields a
knot?
Can we strengthen the conjecture so that the DN that don't satisfy the
pairing condition, but satisfy some other condition always give us
knots?
Is there an example of a DN that does not yield a knot which satisfies
the pairing condition?
We also discussed
composite knots. How do you recognize composite knots in DN?
Monday,
27 October
Work on projects.
Friday,
24 October
Work on projects.
Wednesday,
22 October
Test Blake's conjecture: If each set of even/odd pairs within a Dowker
Notation can be matched consecutively with at least one other pair,
then the Dowker Notation yields a knot.
Monday,
20 October
More on Dowker
notation.
RII moves are noticed
by a negative even number and two pairs of consecutive numbers.
Prime and composite knots.
Homework: Which Dowker
notations give knots?
Do RII moves in Dowker
notations occur whenever there are an odd number of evens in the
notation?
Friday,
17 October
More about Dowker notation.
Which notations give knots?
Homework: Is
your knot chiral or amphichiral?
How do you detect a R-II move in Dowker notation?
Wednesday,
15 October
Dowker notation.
Detecting R-I moves in Dowker notation.
Homework:
Find all Dowker notations for your knot.
Practice drawing your knot from Dowker notations.
Monday,
13 October
Exams returned.
Projects discussed.
Friday,
10 October
Mid-term exam!!!!!
Review sheet
Practice Exam (notice
that I will give you more room on the actual exam
to do your work!)
Wednesday,
8 October
Review for Exam.
Monday,
6 October
More discussion of unknotting
number and p-colorings.
Friday,
3 October
Unknotting number
Homework:
Find the unknotting number of your knot.
Help the stuck member of your group find a p for which their knot is
p-colorable.
Wednesday, 1 October
p-colorability.
Homework:
Finish finding a prime number p so that your knot is p-colorable.
Monday, 29 September
Modular arithmetic.
p-colorability.
Homework:
Find a prime number p so that your knot is p-colorable.
Friday, 26 September
Discussion of: the way to
demonstrate that
tricolorability is not affected by R3 moves. There are 4
different styles of R3 moves and many ways to do the initial
tricoloring. Make sure that tricolorability is preserved in all cases.
Wednesday, 24 September
Tricolorability is not affected
by R1 or R2 moves. If it is also not affected by R3 moves, it
is truly an invariant!
Homework:
Show that
tricolorability is not affected by R3 moves. There are 4
different styles of R3 moves and many ways to do the initial
tricoloring. Make sure that tricolorability is preserved in all cases.
Wednesday,
10 September
More about tricolorability.
Reidemeister moves.
Homework:
Use Reidemeister moves to (very carefully) show that the Monster is the unknot.
Monday, 8 September
Tricolorability.
Homework:
Does the number of crossings correspond to the number of strands of the
knot projection?
(except for in the standard projection of the unknot, of course.)
Is your pet knot tricolorable?
Friday, 5 September
The number of 4
crossing knots.
The number of knots of
each crossing number.
Friday, 5 September
The number of 4
crossing knots.
The number of knots of
each crossing number.
Wednesday, 3
September
Homework collected
Friday, 29 August
There are no one crossing knots
There are no two crossing knots
There is one (pair of) three crossing knots
Homework: How many four crossing knots are there?
Can you find a projection of the trefoil knot with 4 crossings? with 5
crossings? with 13 crossings?
Wednesday, 27 August
Pet Knots chosen:
Grady: 9_7
Blake: 8_1
Nick: 8_18
Anthony: 7_3
Michelle: 8_19
Maegan: 7_7
Tammy: 6_2
Amanda: 7_1
Tracy: 7_4
LeAnn: 8_5
Colleen: 9_23
Tiffany: 9_10
Brittany: 8_16
Sarah: 9_3
Lindsey: 9_1
Thomas: 7_5
Nina: 8_2
Knot projections.
Alternating knots.
The Monster is the is
unknot.
Crossing number.
Homework:
How many one-crossing knots are there?
How many two-crossing knots are there?
Monday, 25 August
Syllabus
Homework:
Pick
a pet knot for Wednesday.
