Homework 5

This homework is due on Thursday, April 15, right before class.

• Alex: 5.5/12, 5.5/18, 5.5/20, 5.5/35, 5.5/33, 10) below,
• Amilcar: 5.5/4, 5.5/11, 5.5/18, 5.5/20, 5.5/28, 1) below, 5) below,
• Bobby: 5.5/4, 5.5/11, 5.5/19, 5.5/22, 5.5/29, 3) below, 8) below,
• Caitlin: 5.5/3, 5.5/13, 5.5/19, 5.5/20, 5.5/28, 4) below, 5) below,
• Callie: 5.5/12, 5.5/18, 5.5/20, 5.5/35, 5.5/33, 11) below,
• Carlos: 5.5/12, 5.5/18, 5.5/20, 5.5/30, 5.5/33, 12) below,
• Colin: 5.5/13, 5.5/18, 5.5/28, 5.5/35, 5.5/33, 10) below,
• Diego: 5.5/4, 5.5/13, 5.5/19, 5.5/20, 5.5/29, 1) below, 6) below,
• Frank: 5.5/4, 5.5/13, 5.5/19, 5.5/20, 5.5/29, 2) below, 8) below,
• Jessica: 5.5/3, 5.5/13, 5.5/18, 5.5/20, 5.5/31, 2) below, 9) below,
• Jonathan: 5.5/4, 5.5/13, 5.5/18, 5.5/22, 5.5/31, 1) below, 9) below,
• Jordan: 5.5/12, 5.5/18, 5.5/20, 5.5/35, 5.5/33, 12) below,
• Lacie: 5.5/11, 5.5/19, 5.5/20, 5.5/35, 5.5/33, 9) below,
• Madison: 5.5/4, 5.5/12, 5.5/19, 5.5/21, 5.5/28, 4) below, 6) below,
• Milani: 5.5/3, 5.5/13, 5.5/19, 5.5/20, 5.5/28, 3) below, 5) below,
• Mikhail: 5.5/4, 5.5/13, 5.5/19, 5.5/20, 5.5/28, 3) below, 7) below,
• Mounia: 5.5/3, 5.5/11, 5.5/18, 5.5/22, 5.5/30, 2) below, 7) below,
• Nada: 5.5/3, 5.5/13, 5.5/18, 5.5/22, 5.5/28, 4) below, 8) below,
• Rozie: 5.5/12, 5.5/19, 5.5/28, 5.5/35, 5.5/33, 9) below,
• Salem: 5.5/4, 5.5/13, 5.5/19, 5.5/20, 5.5/28, 3) below, 7) below,
• Sarah: 5.5/4, 5.5/12, 5.5/18, 5.5/21, 5.5/28, 2) below, 6) below,
• Sonja: 5.5/12, 5.5/18, 5.5/28, 5.5/35, 5.5/33, 10) below,
• Stephen: 5.5/11, 5.5/19, 5.5/20, 5.5/30, 5.5/33, 11) below,
• Verena: 5.5/3, 5.5/11, 5.5/18, 5.5/21, 5.5/31, 1) below, 7) below,

2) Which one of the following two knot projections is alternating? Based on this, can you tell whether any of them might be equivalent to the trefoil knot?

3) Which one of the following two knot projections is alternating? Based on this, can you tell whether any of them might be equivalent to the figure eight knot?

4) Which one of the following two knot projections is alternating? Based on this, can you tell whether they might be equivalent?

5) Show that the following can be extended into a valid 11-coloring of the knot?

6) Show that the following can be extended into a valid 3-coloring of the knot. Then give one of two possible reasons why this knot is not equivalent to the figure eight knot.

7) Can the following be extended into a valid 3-coloring of the knot? If it can, show how. If it cannot, show why not. Is this knot equivalent to the trefoil? Is it equivalent to the figure-eight knot? Are there any hard reasons for your conclusion?

8) Can the following be extended into a valid 5-coloring of the knot? If it can, show how. If it cannot, show why not.

9) Can the following be extended into a valid 5-coloring of the knot. Does this imply that this knot is not 5-colorable? Is it 5-colorable? Then give one of two possible reasons why this knot is not equivalent to the figure eight knot.

11) Is any of the two knots below 3-colorable? Is any of them 5-colorable? Based on this, can you draw any conclusion whether the two knots are equivalent or not? Could you also use the Theorem on alternating knots to back up your claim?

12) Is any of the two knots below 3-colorable? Is any of them 5-colorable? Based on this, can you draw any conclusion whether the two knots are equivalent or not? Could you also use the Theorem on alternating knots to back up your claim?