A Really Great Presentation... or Knot Chloe Avery, Talon Stark, - - PowerPoint PPT Presentation

A Really Great Presentation... or Knot

Chloe Avery, Talon Stark, Xiaoyu Qiao, and Jerry Luo

University of California, Santa Barbara

March 14, 2015

What is a Knot?

A mathematical knot is a closed path in 3-dimensional space that doesn’t self-intersect.

Knot Diagrams

In two dimensional space, we represent knots in the following way:

The Unknot

The unknot is a knot that can be untangled to become a loop with no crossings.

Complicated Unknot

It can be remarkably difficult to determine if a given knot is actually the unknot:

Crossing Game

One interesting thing we can do with knots is play games on them! To do this, take any knot:

Crossing Game

Suppose you “forget” what happens at all of the crossings: We will use this picture as our game board.

Crossing Game

Players take turns making one of the following moves:

Crossing Game

On our diagram, this looks like the following: r

Crossing Game

There are two players:

◮ Player U

Objective: Create an unknot.

◮ Player K

Objective: Create a knot that isn’t an unknot.

Example

Winning Strategy for 31

Does Player K have a winning strategy?

Winning Strategy for 31

Frayed Knot.

Winning Strategy for 31

Due to symmetry, the first move is arbitrary.

Winning Strategy for 31

Also due to symmetry, there are only two possibilities for the second move.

Winning Strategy for 31

This gives us four possibilities for the third move.

Winning Strategy for 31

Recall that for Player U to win, the game must result in an unknot and for Player K to win, the game must result in a knot (in this case the trefoil).

Winning Strategy for 31

Let’s start by assuming that Player K (in red) goes first: On the second move, Player U (in blue) can simply pick the move

• n the left and force Player K to make a move that creates the

unknot.

Winning Strategy for 31

So if Player K goes first, Player U has a winning strategy.

Winning Strategy for 31

Now, let’s consider the case where Player U (in blue) goes first: Regardless of the move that Player K makes in the second move, Player U is always able to make a move that creates the unknot.

Winning Strategy for 31

So if Player U goes first, Player U has a winning strategy.

Winning Strategy for 31

Therefore, we have shown the following: regardless of who goes first, Player U has a winning strategy.

Braid Knots

Another kind of knot that we looked at is called a braid knot:

Winning Strategy for Braid Knots

Claim: Player U wins for Braid Knots. (Note: Braid Knots have an odd number of crossings.)

Winning Strategy for Braid Knots

Examples of Braid Knot Diagrams:

Winning Strategy for Braid Knots

Observation: when one strand goes over the other twice, it reduces to a braid knot with 2 fewer crossings.

Winning Strategy for Braid Knots

Observation: When one strand goes over the other twice, it reduces to a braid knot with 2 fewer crossings.

Winning Strategy for Braid Knots

Therefore, Player U can continue to force this until the braid knot becomes the trefoil: We know that Player U has a winning strategy for the trefoil. So U wins!

Winning Strategy for Odd Tri-Braid Knots

Consider the following kind of knot, called an “Odd-Tri-Braid” knot:

Winning Strategy for Odd Tri-Braid Knots

Winning Strategy for Odd Tri-Braid Knots

Winning Strategy for Odd Tri-Braid Knots

With the exception of one move in Region 1, Player U can always make a strand go over another twice. This reduces that region by 2 crossings.

Winning Strategy for Odd Tri-Braid Knots

Player U can continue to reduce the knot until there is a single move without a match for reduction. We have two possibilities: In either case, Player U has a winning strategy.

Future Work

Winning Strategy for 41

The second player has a winning strategy on the following knot:

Future Work

We are looking into finding a general strategy for knots like these:

Winning Strategy for 74

The first player has a winning strategy on the following knot:

Future Work

We are looking into finding a general strategy for knots like these:

Questions?

References

◮ Winning Ways for your Mathematical Plays by: Elwyn R.

Berlekamp, John H. Conway, and Richard K. Guy

◮ www.math.washington.edu/ morrow/papers/will − thesis.pdf ◮ faculty.smcm.edu/sganzell/papers/untangle − 2.pdf ◮ www.csuchico.edu/math/mattman/NSF/Lecturenotes.pdf ◮ www.math.washington.edu/ ∼

mathcircle/mathhour/talks 2014/henrich − slides.pdf

◮ The Knot Book by Colin C. Adams ◮ Knots, Links, Braids and 3-Manifolds: An Introduction

to Low-Dimensional Topology by V.V. Prasolov and A.B. Stossinsky

◮ An Introduction to Knot Theory by W.B. Raymond

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