Knot Theory
Katie Jacques and Mia Kovan
What is Knot Theory?
Knot theory is the study of closed curves in three dimensions and their possible deformations.
The main question in knot theory is whether two knots are the same, and this question has led to the expansion and application of knot theory.
What is a Knot?
Definition: The projection of a knot is an image of the knot in 2D space
The Knot Equivalence Problem
The Perko Pair
Images: https://www.newscientist.com/article/mg25533950-900-how-many-knots-exist-a-new-computing-trick-is-untangling-the-answer/
Types of Knots
Deformations
Otherwise known as Planar Isotopy, deformations changes the projection of a knot using Reidemeister Moves.
The 3 Reidemeister moves are twisting, poking, and sliding.
Images: https://mathoverflow.net/questions/443054/finite-application-of-one-of-reidemeister-moves-on-a-knot-diagram
Reidemeister Move One: Twist
Twisting the projection of a knot is the creation of a loop in a knot, as shown below.
Images: https://www.researchgate.net/figure/The-three-Reidemeister-moves-Each-move-corresponds-to-the-simplest-changes-in-a-diagram_fig1_338149491
Reidemeister Move Two: Poke
Poking is the act of taking one section of a knot and placing it under or over another section.
Images: https://www.researchgate.net/figure/The-three-Reidemeister-moves-Each-move-corresponds-to-the-simplest-changes-in-a-diagram_fig1_338149491
Reidemeister Move Three: Slide
Sliding is when you take one section of a knot and slide it so that it is on the other side of a crossing of two other sections.
Images: https://www.researchgate.net/figure/The-three-Reidemeister-moves-Each-move-corresponds-to-the-simplest-changes-in-a-diagram_fig1_338149491
Tricolorability
Tricolorability is the ability for a knot to be colored in 3 colors where crossings have to be either all the same color or all 3 colors, and at least 2 colors must be used.
Reidemeister Moves and Tricolorability
Using Reidemeister moves changes the ways that knots can be colored, as demonstrated in the image below.
Images: https://ima.org.uk/17434/whats-knot-to-love/
Invariants
An invariant is a property of a knot that can be used to determined if it is different from another knot
BUT: cannot tell us for sure whether two knots are equivalent!!
Examples of Invariants
Crossing Number
Crossing number is one of the most obvious invariants, but has an important caveat. The crossing number of a knot is defined as the amount of crossings in the simplest projection of the knot, and so doesn’t work for any projection.
Tricolorability
Tricolorability allows us to differentiate between knots that are tricolorable and knots that are not. A knot that is tricolorable cannot be the same knot as a knot which is not tricolorable.
P-Colorability
P-colorability is the general version of Tricolorability. It numbers strands from 0 to p-1, and again must use at least two numbers. In order for a knot to be p-colorable, the following equation must be true:
(b1+b2)(modp)=2t(modp)
where b1 and b2 are the numbers of the bottom strand and t is the number of the top strand.
Alexander Polynomial
The Alexander Polynomial was developed in 1923, and is an invariant which allows for more differentiation between knots.
It tells us that A(O)=1, and then
A(frw)-A(bkw)+(t1/2-t-1/2)A(sep)=0
Other Polynomial Invariants
Jones polynomial: Discovered in 1984, the Jones polynomial uses additional variables and therefore distinguishes between more knots.
HOMFLY-PT polynomial: Uses multiple variables in order to further characterize knots.
How to distinguish knots
Invariants allow us to tell when two knots are not equal, but just because two knots have the same invariant properties does not mean they are the same. This is why the Knot Equivalence Problem remains unanswered to this day.
Reidemeister moves allow us to attempt to deform one knot into another, proving equivalence, but there is no theorem which provides a general equivalence.
Braids
A braid is a collection of strands connecting two rows of points, where strands must begin and end at two parallel planes, but can move in the middle.
Some Theorems about Braids
Alexander’s Theorem states that every knot or link can be represented as a closed braid.
Markov’s Theorem states that equivalent braids expressing the same link are connected by applications of two types of Markov moves.
First off: how do I stop my wires from knotting
Study conducted in 2007 by jostling strings around in an enclosed space
So we should all be keeping our wires in a small enclosed space to prevent tangling
Application: molecular knotting
Proteins have knots in them
Which knots? How do we extract these knots?
Protein structure
Amino acids, each of which has an N-C-C structure
Protein structure
Amino acids, each of which has an N-C-C structure
Protein structure
Backbone gets twisted and folded up
Where are the knots
=
?
Finding the knots
First issue: making a closed loop
Finding the knots
First issue: making a closed loop
terminus 1
terminus 2
Finding the knots
First issue: making a closed loop
How do we connect the termini without accidentally creating or untying knots?
terminus 1
terminus 2
Finding the knots
Smoothing algorithm by Taylor (2000)
So what knot actually exist in proteins
Only a few types found thus far
Using this particular smoothing algorithm, Taylor analyzed a selection of 3,440 proteins, all of different structure
5 right-handed trefoils 1 left-handed trefoil 2 figure eights
So what knot actually exist in proteins
Only a few types found thus far
Using this particular smoothing algorithm, Taylor analyzed a selection of 3,440 proteins, all of different structure
5 right-handed trefoils 1 left-handed trefoil 2 figure eights
Images: https://commons.wikimedia.org/wiki/File:The_two_trefoil_knots.pdf, https://www.researchgate.net/figure/The-figure-eight-knot-inside-M-S-3_fig2_282906308
https://en.wikipedia.org/wiki/Stevedore_knot_%28mathematics%29, https://en.wikipedia.org/wiki/Three-twist_knot
POP QUIZ
So what knot actually exist in proteins
Only a few types found thus far
Using this particular smoothing algorithm, Taylor analyzed a selection of 3,440 proteins, all of different structure
5 right-handed trefoils 1 left-handed trefoil 2 figure eights
Images: https://commons.wikimedia.org/wiki/File:The_two_trefoil_knots.pdf, https://www.researchgate.net/figure/The-figure-eight-knot-inside-M-S-3_fig2_282906308
https://en.wikipedia.org/wiki/Stevedore_knot_%28mathematics%29, https://en.wikipedia.org/wiki/Three-twist_knot
POP QUIZ
Are there the same knot? How do you know?
So what knot actually exist in proteins
Only a few types found thus far
Using this particular smoothing algorithm, Taylor analyzed a selection of 3,440 proteins, all of different structure
5 right-handed trefoils 1 left-handed trefoil 2 figure eights
Images: https://commons.wikimedia.org/wiki/File:The_two_trefoil_knots.pdf, https://www.researchgate.net/figure/The-figure-eight-knot-inside-M-S-3_fig2_282906308
https://en.wikipedia.org/wiki/Stevedore_knot_%28mathematics%29, https://en.wikipedia.org/wiki/Three-twist_knot
POP QUIZ
Are there the same knot? How do you know?
What might knots tell us about proteins?
Kind of mysterious how and why they exist
This idea of stability also motivates studying knots in other types of molecules
Knots in other molecules
When our molecule is knotted, it is often trapped in a heightened energy state
Knots in other molecules
When our molecule is knotted, it is often trapped in a heightened energy state
trefoil!
Knotty molecule properties
Knot properties affect molecular properties
Mirror =/= original → Euclidean chiral
Detour: chirality example
Mirror =/= original → Euclidean chiral
Knotty molecule properties
Knot properties affect molecular properties
Mirror =/= original → Euclidean chiral
Images: https://www.catenane.net/pages/2016_knot_catalysis.html, https://en.wikipedia.org/wiki/Cinquefoil_knot
POP QUIZ
Images: https://www.catenane.net/pages/2016_knot_catalysis.html, https://en.wikipedia.org/wiki/Cinquefoil_knot
POP QUIZ
Is this chiral?
Application: why do we live in 3D?
Why 3 dimensions?
Application: why do we live in 3D?
Why 3 dimensions?
Cosmic inflation: theory of expansion of the early universe
knots
POP QUIZ
POP QUIZ
What knots do we see here
So… what about the 3D thing
No nontrivial knot exists in 4D
So… what about the 3D thing
No nontrivial knot exists in 4D
So… what about the 3D thing
No nontrivial knot exists in 4D
So… what about the 3D thing
No nontrivial knot exists in 4D
References:
A deeply knotted protein structure and how it might fold
Knotted proteins: a tangled tale of structural biology
Highly conductive topologically chiral molecular knots as efficient spin filters
Spontaneous knotting of an agitated string
Origins of the universe: Inflation
Filling the early universe with knots can explain why the world is three dimensional
References
Application: Quantum computing
New way of performing computations
But so far, just theory
Qubit??
State of a single qubit is the quantum superposition of |0〉and |1〉with amplitudes α0, α1
When we measure the qubit, there is probability |α0|^2 of observing |0〉and probability |α1|^2 of observing |1〉
As we add qubits, our possibilities increase beyond just |0〉and |1〉
How is this helpful though
More operations and reversibility
TLDR: qubits hold a lot more manipulatable info than cbits do
Knots!!! (specifically braids)
Quantum resources: