Tricolorability and Reidemeister Moves

Hello, again!


This is the second part of our weekly update, and a continuation of our earlier discussion of tricolorability.

So, essentially, we posed the question of -- can a knot still be tricolorable if you twist it around, poke at it, etc? Why does this matter, you ask.

Well, do you remember how we looked at the unknot last time and pointed out that it can't be tricolorable, because there's no way to color the unknot with at least two colors?


Well, it turns out, if you take the unknot and use a few Reidemeister moves on it, you get this

Figure 1 - The unknot with multiple Reidemeister moves. 

What's to say, then, that one of these later arrangements can't be tricolorable.


Thus, it's useful for us to show that the Reidemeister moves do not affect tricolorability.

Below, then, we can see that each of the Reidemeister moves are not affected by tricolorability
(ie - they remain colorable in at least two colors, and at each of the crossings, have all same or all different colored strands):

Note that, below, as each of the Reidemeister moves are acted upon the section of the knot, the tricolorability (all three colors or only one color) is preserved.

Figure 2a - Reidemeister Move 1

Figure 2b - Reidemeister Move II

Figure 2c - Reidemeister Move III.

Tricolorability, in comparison to other invariants, it is slightly easier to deal with, because if a knot is tricolorable in one projection, then it is tricolorable in any other equivalent projection (i.e. - If you stretch/move/twist it around a lot, it's still tricolorable.)

Thanks for listening, and keep reading for the next post on petal numbers and ubercrossings!

References:

"Figure Eight Knot—Dave Richeson - Math 201: Knot Theory". Math201s09.wikidot.com. N.p., 2016. Web. 5 Mar. 2016.

"Knots | Brilliant Math & Science Wiki". Brilliant.org. N.p., 2016. Web. 5 Mar. 2016.

"The Pretzel Knot - Math 201: Knot Theory". Math201s09.wikidot.com. N.p., 2016. Web. 5 Mar. 2016.

"Tricolorable -- From Wolfram Mathworld". Mathworld.wolfram.com. N.p., 2016. Web. 5 Mar. 2016.

"Unknot". Personal.kenyon.edu. N.p., 2016. Web. 5 Mar. 2016.

Adams, Colin Conrad. The Knot Book. New York: W.H. Freeman, 1994. Print.

Nizami, Abdul Rauf, Mobeen Munir, and Malka Shah Bano. "The Quantum Sl≪Sub≫2≪/Sub≫-Invariant Of A Family Of Knots". AM 05.01 (2014): 70-78. Web. 5 Mar. 2016.