The Miracle of Knot

1.Knot Theory 2.Tricolorability

Knot Theory

• A mathematician's knot, formally speaking, is a “closed loop” in R3 . It is a line that we can draw in the space R3 which does not intersect itself and go back to where it started.

• Two mathematical knots are considered the same if one can be “bended” into the other knot in R3. The way of bending should not involve cutting the line or passing the line through itself.

Different Knots

Unknot

• This is important to know that the simplest knot is the unknot, because people often want to figure out about when a knot is the unknot.

• In the definition, we talked about knots in R3, however people usually work on knots in R2, such as what has been shown in previous pictures.

• It’s because working on knots in R3 is much more difficult, and we do it as how we draw a cube in R2. Solid lines demonstrate they are in the front, and lines that are “cut” are in the back.

Is a Given Knot the Unknot?

Reidemeister moves

Reidemeister moves are a set of ways that we can re-draw parts of a knot in R2 by not changing the knot.

• (1) Twist and untwist in either direction.• (2) Move one line completely over another.• (3) Move a line completely over or under a

crossing.

Reidemeister moves

Reidemeister moves on a knot.

One of the Ways to Define a Knot

• Tricolorability• In the mathematical field of knot theory,

the tricolorability of a knot is the ability of a knot to be colored with three colors according to certain rules.

• Tricolorability will not be changed when we draw a knot in other ways by using Reidemeister moves. In this case, we can know that if one knot can be tricolored and the other cannot, then they must be different knots.

Rules of tricolorability

• A knot is tricolorable if each line of the knot diagram can be colored one of three colors, subject to the following rules:

• (1)At least two colors must be used• (2)At each crossing, the three lines that leave

the crossing are either all the same color or all different colors.

Tricolorable or not

YES No

Reidemeister Moves Are Tricolorable.

Twist to Untwist Unpoke to poke

Slide

Conclusion

• A tricolorable knot can’t transfer to an un-tricolorable knot by using reidemeister moves. The opposite way can’t either.

• By using this property, we can know if a given complex knot is tricolorable, then it must not be the unknot.

• The unknot is not tricolorable, any tricolorable knot is necessarily nontrivial.

Ref

• http://en.wikipedia.org/wiki/Knot_theory• http://en.wikipedia.org/wiki/Tricolorability• http://mit.uvt.rnu.tn/NR/rdonlyres/Mathema

tics/18-304Spring-2006/94568A74-E63B-4C2C-82C2-01C14CFD9CF8/0/jacobs_knots.pdf