An Interactive Introduction to Knot Theory by Inga Johnson

Author:Inga Johnson [Johnson, Inga]
Language: eng
Format: epub
Publisher: Dover Publications

Exercise 5.3.3. Which of the diagrams in Figure 5.1.2 are tricolorable?

As mentioned above, tricolorability is a link invariant. We prove this by showing it is impossible for one diagram of a knot K to be tricolorable while another diagram of the same knot is not tricolorable.

Theorem 5.3.4. If a given diagram of a knot, K, is tricolorable, then every diagram of K is tricolorable.

Theorem 5.3.4 ensures that the following definition makes sense and gives our first example of a knot invariant.

Definition 5.3.5. A knot is called tricolorable if its diagrams are tricolorable.

Exercise 5.3.6. Prove Theorem 5.3.4. (Hint: Show that the tricolorability of a diagram is unchanged by the application of R1, R2, and R3 moves.)

Exercise 5.3.7. Prove that the trefoil knot is not equivalent to the unknot.

Exercise 5.3.8. For which values of n is the torus knot T2,n tricolorable?

Exercise 5.3.9. Explore the tricolorability of pretzel links. Make a conjecture and prove it.

Exercise 5.3.10. Are either of the closed braids in Figure 3.4.5 tricolorable?

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