### Clayton Shonkwiler PRO

Mathematician and artist

Clayton Shonkwiler

MATH 192

A *knot* is a closed loop in space

Peter Guthrie Tait

Plasmid DNA

Alonso–Sarduy, Dietler Lab

EPF Lausanne

Which of these are *knots*? (1 piece)

Which are *links*? (More than 1 piece)

Which are *equivalent*?

Is there a way to assign crossings to this projection to make it the unknot?

Two players: the **K**notter and the **U**nknotter.

**K** and **U** take turns assigning crossings; **K**’s goal is to make the diagram knotted and **U**’s goal is to make it unknotted.

In pairs, decide who is **K** and who is **U**.

Then decide who goes first.

Play the game.

Who won? Did the winner have an advantage?

Play again, switching who goes first. Who won? Did they have an advantage?

Thistlethwaite unknot

Ochiai unknot

**Theorem** [Hass–Lagarias–Pippenger]

Unknottedness is in NP.

**Theorem** [Lackenby]

Unknottedness is in co-NP.

Can unknots be recognized in polynomial time?

**Formal Definitions**

A *knot* is a simple, closed, polygonal curve in space.

A *link* is a collection of pairwise non-intersecting knots.

A two-dimensional picture of a knot is a *diagram*.

We often allow the segments to become very short, so the curve appears to be smooth.

- Build a knot, then draw a good diagram of it.
- Now draw a bad diagram of the same knot. For example, can you make a diagram from which the knot cannot be reconstructed from crossing information? Other diagram weirdness?
- Make a list of Properties to Avoid in diagrams.

Reidemeister moves

Find a sequence of Reidemeister moves from \(K_1\) to \(K_2\).

Hint:

For each knot \(K\) shown below, some assignment of crossings in the projection to the left is equivalent to \(K\). Can you find this crossing assignment? And can you show that a sequence of Reidemeister moves connects the two diagrams?

All (prime) knots with 6 crossings

**Theorem** [Reidemeister]

Two knots \(K_1\) and \(K_2\) are equivalent if and only if any diagram for \(K_1\) and any diagram for \(K_2\) can be related by a sequence of Reidemeister moves.

**The Fundamental Problem of Knot Theory**

How can you tell when two knots are *not* equivalent?

- \(>1.8\) trillion random knots generated
- 92.8% unknots
- 13,020 distinct knot types, including 20 different 19-crossing knots
- \(\approx\)300,000 core-hours of CPU time (or 34.2 core-years)

How can we tell when two knots are *not* equivalent? How to prove it?

How can we tell when two knots are equivalent? How to prove it?

equivalent?

**Definition**

A *knot invariant* is a function whose domain is the set of equivalence classes of knots.

Invariant | Value |
---|---|

Bridge index | 2 |

Alternating | T |

Hyperbolic volume | 2.029883213 |

Alexander polynomial | |

HOMFLY polynomial | |

Knot Floer homology |

1-3t+t^2

\frac{1}{v^2} - 1 + v^2 - z^2

If you suspect two knots are *not* equivalent, find an example of an invariant you can compute, then show that the two knots produce different values.

(2-2v^2+v^4)+(1-3v^2+v^4)z^2-\frac{1}{v^2}z^4

\left(\frac{-1}{v^2}+3-v^2\right)+\left(\frac{-1}{v^2}+3-v^2\right)z^2+z^4

equivalent?

No!

HOMFLY polynomial

**Definition**

A knot diagram is called *tricolorable* if each arc in the diagram can be drawn using one of three colors, say blue, green, and pink, in such a way that the following two conditions hold:

- At least two colors are used in the diagram.
- At each crossing, either all arcs are colored the same or all arcs are colored differently.

Finish assigning colors to the black arcs in this diagram to give a valid tricoloring.

Which of the diagrams in this table are tricolorable?

**Theorem** [Fox]

If any diagram of a knot \(K\) is tricolorable, then *every* diagram of \(K\) is tricolorable.

**Definition**

A knot is called *tricolorable* if any (and hence every) diagram for the knot is tricolorable.

Is the figure-eight knot tricolorable?

Prove that the trefoil and the unknot are not equivalent.

By Clayton Shonkwiler

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