Invariants of Knots

A presentation by Jade Wirz

What is a Knot?

Remember the glow sticks?

Knot Diagrams

Reidemeister Moves

• Complete list of moves
• D₁ ≣ D₂ via a series of these moves if and only if K₁ ~ K₂
• Proven Independently by Kurt Reidemeister and James Waddell Alexander

Some Equivalent Diagrams

Boolean Invariant

Colorability

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Rules:

There are n distinct colors colors

Color each “arc” of the knot diagram

Each intersection must either be all the same or all different colors

You must use each color at least once

The Trefoil is 3-colorable

The unknot is not 3-colorable

Integer Invariants

Oriented Knots

We add an orientation on the knot, (we put arrows everywhere on it)

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Writhe

Not actually an invariant!

Is an “invariant” of an oriented knot

At each “intersection” add one if it’s positive, subtract if it’s negative

Positive and Negative Crossing by Convention

Each of the above crossings are positive, so this knot has a writhe of 3

What is the Writhe of this knot diagram?

What is the Writhe of this knot diagram?

What is the Writhe of this knot diagram?

What is the Writhe of this knot diagram?

2 - 2 = 0

Writhe “Invariance” for RMI

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Writhe Invariance for RM II

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Writhe Invariance for RMIII

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Polynomial Invariants

Kauffman Bracket Relation

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• Another “Invariant”
• Notation
• L is a diagram
• O is the unknot
• 〈L〉is the Laurent polynomial of L
• More rules (recursively defined by)

Is neither a knot, or a diagram, but in fact some secret, third thing

Behavior Under RM1

Not an invariant under RM1

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Let’s Work Out RM 2 and 3 Together!

The Jones Polynomial

Idea: Combine Writhe and The Bracket Polynomial in such a way that cancels out!

A “positive” twist = -A^-3.

(-A)^-3w(L)〈L〉

Is an invariant under RM1, RM2, RM3, and therefore is an invariant of knot diagrams

By convention, we substitute A^-2 = t^1/2.

Let’s compute an example!

Let’s compute an example!

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A

A^-1

A²

1

A^-2

1

A³

A

A

A^-1

A

A^-1

A^-1

A^-3

Let’s compute an example!

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simplifying gives us

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-A⁵-A^-3+A^-7

Let’s compute an example!

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(-A)^-3w(D)(-A⁵-A^-3+A^-7)

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Now we substitute A^-4 = t

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J_trefoil(t) = t+t³-t⁴

Let’s try the Jones Polynomial!

Let’s try the Jones Polynomial!

Let’s try the Jones Polynomial!

Introducing: The Colored Jones Polynomial

Boxes

Boxes

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Algebra and the Annuli

The 2^nd Chebyshev polynomial

2^nd Chebyshev Polynomial

General form of Δ_n

The Annuli

Cabling a Knot

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Computing an Example!

Possible Cablings of the Hopf Link

n = 2

k = Hopf Link

What We’ve Covered

• Knots and Knot Diagrams
• Knot Invariants
• Connected Knots to 3-Manifolds
• Colored Jones Polynomial
• Temperley Lieb Algebra
• Found an invariant of Knot Complements

What to Learn Next:

• Choices for values of A (from the Jones Polynomials)
• Quantum Integers
• Topological Cryptography
• Hyperbolic/Spherical/Euclidean Space of Knot Complements
• Volumes of Knots

Thank You!