the algebra of knots - fullerton collegestaffsam nelson the algebra of knots. why knots? many...

The Algebra of Knots

Sam Nelson

Claremont McKenna College

Sam Nelson The Algebra of Knots

Familiar Operations and Sets

Addition comes from unions:

Multiplication comes from matched pairs:

Algebraic Properties

The properties of these familiar operations reflect properties ofthe set operations which inspired them.

Example: Associativity of addition

Example: Commutativity of multiplication

Why sets?

Many useful real-world quantities behave like sets

Examples: lengths, masses, volumes, money

However, not every quantity behaves so simply

Examples: waves interfere, particles become entangled

Let us now consider operations inspired by knots:

Why sets?

Definition: A knot is a simple closed curve inthree-dimensional space.

simple - does not intersect itself

closed - has no loose endpoints, i.e. forms a loop

Knot Diagrams

We represent knots with pictures called knot diagrams.

Knot Diagrams

Drawing a knot from a different angle or moving it around inspace will result in different diagrams of the same knot.

Topology

Knots are topological objects, meaning that we consider twoknots the same if one can be changed into the other by:

moving the knot in space

stretching or shrinking in a continuous way

but not cutting and retying

Topology

Why Knots?

Many molecules (polymers, protein, DNA) are knots, andtheir chemical properties are determined in part by howthey’re knotted

Certain antibiotics work by blocking the action ofmolecules called topoisomerase which change how DNA isknotted; blocking the unknotting of the DNA stops thebacteria reproducing

Perhaps surprisingly, the mathematics of knots is relevantto the search for a theory of quantum gravity, a majorunsolved problem in physics

Besides, knots are just fun!

Why Knots?

The main problem

Given two knot diagrams K and K ′, how can we tell whetherthey represent the same knot?

Reidemeister Moves

Example

Knot Invariants

Quantities we can compute from a knot diagram

Get the same value for all diagrams of the same knot

Unchanged by Reidemeister moves

If K and K ′ have different invariant values, they representdifferent knots

Knot Invariants

Examples:

Alexander/Jones/HOMFLYpt/Kauffman polynomials

Knot group

Hyperbolic Volume

Khovanov Homology

Knot Invariants

Examples:

Knot group

Hyperbolic Volume

Khovanov Homology

Knot Invariants

Examples:

Knot group

Hyperbolic Volume

Khovanov Homology

Knot Invariants

Examples:

Knot group

Hyperbolic Volume

Khovanov Homology

Knot Invariants

Examples:

Knot group

Hyperbolic Volume

Khovanov Homology

Knot Invariants

Examples:

Knot group

Hyperbolic Volume

Khovanov Homology

An algebraic knot invariant

Define an algebraic operation . from knot diagrams

Use Reidemeister moves to determine properties of theoperation

Find operations satisfying these axioms either by combiningexisting operations or by making operation tables

(a.k.a. Involutory Quandles)

Attach a label to each arc in a knot diagram

When x goes under y, the result is x . y

Kei Axioms

x . x = x

Kei Axioms

(x . y) . y = x

Kei Axioms

(x . y) . z = (x . z) . (y . z)

Thus, a kei is a set of labels with an operation . such that forall x, y, z we have

(i) x . x = x

(ii) (x . y) . y = x

(iii) (x . y) . z = (x . z) . (y . z)

(i) x . x = x

(ii) (x . y) . y = x

(iii) (x . y) . z = (x . z) . (y . z)

(i) x . x = x

(ii) (x . y) . y = x

(iii) (x . y) . z = (x . z) . (y . z)

(i) x . x = x

(ii) (x . y) . y = x

(iii) (x . y) . z = (x . z) . (y . z)

Kei Example

We can define kei operations using existing operations. Forexample, the integers Z have a kei operation given by

x . y = 2y − x.

To see that this is a kei operation, we need to verify that itsatisfies the axioms. For example,

x . x = 2x− x = x

Kei Example

x . y = 2y − x.

x . x = 2x− x = x

Kei Example

x . y = 2y − x.

x . x = 2x− x = x

Kei Example

We can also specify a kei operation with an operationtable, just like a multiplication table:

. 1 2 3

1 1 3 22 3 2 13 2 1 3

Checking the axioms here must be done case-by-case and isbest handled by computer code.

Kei Example

. 1 2 3

1 1 3 22 3 2 13 2 1 3

Kei Example

. 1 2 3

1 1 3 22 3 2 13 2 1 3

The counting invariant

A valid labeling of a knot diagram by a kei must satisfy thecrossing condition at every crossing.

. 1 2 3

1 1 3 22 3 2 13 2 1 3

If a labeling of a knot diagram by a kei fails to satisfy thecrossing condition at any crossing, it is invalid.

. 1 2 3

1 1 3 22 3 2 13 2 1 3

Because of the kei axioms, every valid labeling of a diagrambefore a move corresponds to a unique valid labeling after amove.

Any two diagrams of the same knot will have same number ofvalid labelings by your favorite kei. So, to tell knots apart,

we count the valid labelings!

Any two diagrams of the same knot will have same number ofvalid labelings by your favorite kei. So, to tell knots apart,

we count the valid labelings!

If our set of labels is finite, there are only finitely manypossible labelings

So we can list all possible labelings and count how manyare valid

This number will be the same for all diagrams of K

So it is a knot invariant, called the kei counting invariant

The unknot has three labelings by our three-element kei:

A knot invariant

The trefoil has nine labelings by our three-element kei:

A knot invariant

3 6= 9, and the kei counting invariant shows that no sequence ofReidemeister moves can unknot the trefoil.

Generalizations of Kei

An oriented knot has a preferred direction of travel.

Orienting K changes axiom (ii) to allow .−1 to be differentfrom .; the resulting object is called a quandle

Replacing the type I move with a doubled version gives usframed knots which are like physical knots

This eliminates quandle axiom (i); the resulting object iscalled a rack

Dividing the arcs at over-crossings to get semi-arcs letsboth inputs act on each other

The resulting algebraic structures are called bikei,biquandles and biracks

These are solutions to the Yang-Baxter Equation fromstatistical mechanics

Enhanced Invariants

A signature is a quantity computable from a labeled knotdiagram which is invariant under labeled moves

Counting signatures instead of labelings defines strongerinvariants

Enhanced Invariants

Example: For each labeling, count uc where c is the number oflabels used. This is called the image-enhanced countinginvariant, denoted ΦIm

Enhanced Invariants

ΦImX (Unknot) = u + u + u = 3u

Enhanced Invariants

ΦImX (31) = 3u + 6u3

Enhancements

Enhancements can come from the general structure of thelabeling object.

Image enhancements

Writhe enhancements

Quandle polynomials

Column groups

Enhancements

Image enhancements

Writhe enhancements

Quandle polynomials

Column groups

Enhancements

Image enhancements

Writhe enhancements

Quandle polynomials

Column groups

Enhancements

Image enhancements

Writhe enhancements

Quandle polynomials

Column groups

Enhancements

Image enhancements

Writhe enhancements

Quandle polynomials

Column groups

Enhancements

Enhancements can come from using special types of labelingobjects.

Symplectic quandle invariants

Bilinear biquandle invariants

Coxeter rack invariants

(t, s)-rack invariants

Enhancements

Enhancements can also come from additional structures.

Quandle 2-cocycles

Rack shadows

Rack algebras

Enhancements

Quandle 2-cocycles

Rack shadows

Rack algebras

Enhancements

Quandle 2-cocycles

Rack shadows

Rack algebras

Enhancements

Quandle 2-cocycles

Rack shadows

Rack algebras

Tying it all up...

Algebra from knots seems weird at first,

but the algebraic structures arising from knots haveapplication in biology, chemistry, physics. . .

and who knows where else?

You can help find out!

Tying it all up...

Thanks for Listening!

the algebra of knots - fullerton collegestaffsam nelson the algebra of knots. why knots? many...

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