Clayton Shonkwiler PRO
Mathematician and artist
Knot Theory
Clayton Shonkwiler
MATH 192
A knot is a closed loop in space
Peter Guthrie Tait
Plasmid DNA
Alonso–Sarduy, Dietler Lab
EPF Lausanne
Vortex Knots
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?
The Basic Problems
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?
Knot Invariants
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
Ansatz
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
How hard is it to recognize unknots?
Theorem [Hass–Lagarias–Pippenger]
Unknottedness is in NP.
Theorem [Lackenby]
Unknottedness is in co-NP.
An Unsolved Problem in Mathematics
Can unknots be recognized in polynomial time?
Thank you!
The Unknotting–Knotting Game
Two players: the Knotter and the Unknotter.
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?
Knot Theory
By Clayton Shonkwiler
