| Topics Covered | Homework |
|---|---|
| Syllabus | Get book Pick your pet knot Come on Wednesday ready to work |
| Topics Covered | Homework |
|---|---|
| What is a knot? What is a link? Crossings in a projection Crossing number of a knot Knots of crossing number 0 Knots of crossing number 1 Knots of crossing number 2 Amphichiral knots | Identify the 4 crossing knot drawn in class
Identify the monster - which knot is it? Do exercise 1.3 |
| Topics Covered | Homework |
|---|---|
| The 4 crossing knot drawn in class was the trefoil! The monster was the unknot! Any knot has a projection with 1000 crossings! Alternating knots Changing crossings to get to the unknot Unknotting number | Find the unknotting number of your knot.
Why can every knot be made into the unknot by changing some number of crossings? |
| Topics Covered | Homework |
|---|---|
|
unknotting numbers of everyone's pet knots
|
Find the unknotting number of the 8_18 knot Find a knot whose unknotting number is unknown Exercise 1.6 Exercise 1.7 - Why can every knot be made into the unknot by changing some number of crossings? |
| Topics Covered | Homework |
|---|---|
|
Unknotting numbers of 8_18 Knots with unknown unknotting number See this page to see that the 8_10, 9_10, 9_13, 9_29, and other knots have unknown unknotting numbers See this page for one example of a knot whose minimal crossing projection does not have the minimal unknotting number Why can every knot be unknotted? |
Read Section 1.2 |
| Topics Covered | Homework |
|---|---|
|
unknotting number revisited one last time prime numbers Ccomposite numbers composite knots prime knots numbers of prime knots |
Find u(8_19) Read the article |
| Topics Covered | Homework |
|---|---|
|
u(8_19) Reidemeister moves |
Find a sequence of R-moves to change the picture in class into its standard presentation Read the article |
| Topics Covered | Homework |
|---|---|
|
Tricoloration It is invariant under Reidemeister moves |
Is your knot tricolorable? Finish checking that tricoloration is invariant under Reidemeister moves Read the article |
| Topics Covered | Homework |
|---|---|
|
Whose knots are Tricolorable? It is invariant under Reidemeister moves Modular arithmetic cAdding in different bases |
Finish checking that tricoloration is invariant under Reidemeister moves Read the article |
| Topics Covered | Homework |
|---|---|
|
Modular arithmetic p-coloration The 4_1 knot is 5-colorable The unknot is not 5-colorable The 4_1 is not the unknot |
Is the monster 5-colorable? (For Monday) Find a prime p (other than 3) so that your knot is p-colorable Read the article |
| Topics Covered | Homework |
|---|---|
|
p-coloration If a knot has a 3-coloration, must it have a 5-coloration? If a knot has a p-coloration, must it have a np-coloration? |
Find a prime p (other than 3) so that your knot is p-colorable Read the article |
| Topics Covered | Homework |
|---|---|
|
Dowker notation Calculating Dowker notation Recreating a knot from Dowker notation |
Find a prime p (other than 3) so that your knot is p-colorable Find a non-prime (other than a multiple of 3) so that your knot is p-colorable (For Fri) Find all possible Dowker notations for your knot Read the article |
| Topics Covered | Homework |
|---|---|
|
Dowker notation Calculating Dowker notation Recreating a knot from Dowker notation Dowker notation for alternating knots |
Find a prime p (other than 3) so that your knot is p-colorable Find a non-prime (other than a multiple of 3) so that your knot is p-colorable (For Fri) Find all possible Dowker notations for your knot Exercise 2.9 Read the article |
| Topics Covered | Homework |
|---|---|
|
How to tell when Dowker notation indicates an Reidemeister Type 1 move |
Find a prime p (other than 3) so that your knot is p-colorable Find a non-prime (other than a multiple of 3) so that your knot is p-colorable Finish Excercise 2.9 Read the article |
| Topics Covered | Homework |
|---|---|
|
How to tell when Dowker notation indicates an Reidemeister Type 2 move |
Find a prime p (other than 3) so that your knot is p-colorable Find a non-prime (other than a multiple of 3) so that your knot is p-colorable Write a random string of Dowker notations and determine if it represents a Type 1 or Type 2 move Read the article |
| Topics Covered | Homework |
|---|---|
|
How to tell when Dowker notation indicates an Reidemeister Type 2 move Testing Josh's conjectures |
Find a prime p (other than 3) so that your knot is p-colorable Find a non-prime (other than a multiple of 3) so that your knot is p-colorable Write a random string of Dowker notations and determine if it represents a Type 1 or Type 2 move Read the article |
| Topics Covered | Homework |
|---|---|
|
Conway Notation |
Do exercise 2.11 determine the conway notation for your knot (if it has one) Read the article |
| Topics Covered | Homework |
|---|---|
|
Conway Notation Discuss exercise 2.11 Continued fractions |
Do exercise 2.13 determine the conway notation for your knot (if it has one) Read the article |
| Topics Covered | Homework |
|---|---|
|
Discuss exercise 2.13 Discuss rational tangle notation for your knots |
determine the conway notation for your knot (if it has one) Read the article |
| Topics Covered | Homework |
|---|---|
|
Rational tangles and their unknotting numbers - see page 62 |
Determine the unknotting number of the presentation 2 -2 2 -2 2 4 Read the article |
| Topics Covered | Homework |
|---|---|
|
Mathematically celtic knots
|
Determine whether your knot is mathematically celtic. Read the article |
| Topics Covered | Homework |
|---|---|
|
Mathematically celtic knots, again
|
Write up whether your knot is mathematically celtic. Are all alternating knots mathematically celtic? Read the article |