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The Algebra of Knots
Sam Nelson
Claremont McKenna College
Sam Nelson The Algebra of Knots
Familiar Operations and Sets
Addition comes from unions:
Sam Nelson The Algebra of Knots
Familiar Operations and Sets
Addition comes from unions:
Sam Nelson The Algebra of Knots
Familiar Operations and Sets
Multiplication comes from matched pairs:
Sam Nelson The Algebra of Knots
Familiar Operations and Sets
Multiplication comes from matched pairs:
Sam Nelson The Algebra of Knots
Algebraic Properties
The properties of these familiar operations reflect properties ofthe set operations which inspired them.
Sam Nelson The Algebra of Knots
Algebraic Properties
Example: Associativity of addition
Sam Nelson The Algebra of Knots
Algebraic Properties
Example: Associativity of addition
Sam Nelson The Algebra of Knots
Algebraic Properties
Example: Associativity of addition
Sam Nelson The Algebra of Knots
Algebraic Properties
Example: Commutativity of multiplication
Sam Nelson The Algebra of Knots
Algebraic Properties
Example: Commutativity of multiplication
Sam Nelson The Algebra of Knots
Algebraic Properties
Example: Commutativity of multiplication
Sam Nelson The Algebra of Knots
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:
Sam Nelson The Algebra of Knots
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:
Sam Nelson The Algebra of Knots
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:
Sam Nelson The Algebra of Knots
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:
Sam Nelson The Algebra of Knots
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:
Sam Nelson The Algebra of Knots
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:
Sam Nelson The Algebra of Knots
Knots
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
Sam Nelson The Algebra of Knots
Knots
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
Sam Nelson The Algebra of Knots
Knots
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
Sam Nelson The Algebra of Knots
Knot Diagrams
We represent knots with pictures called knot diagrams.
Sam Nelson The Algebra of Knots
Knot Diagrams
Drawing a knot from a different angle or moving it around inspace will result in different diagrams of the same knot.
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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!
Sam Nelson The Algebra of Knots
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!
Sam Nelson The Algebra of Knots
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!
Sam Nelson The Algebra of Knots
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!
Sam Nelson The Algebra of Knots
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!
Sam Nelson The Algebra of Knots
The main problem
Given two knot diagrams K and K ′, how can we tell whetherthey represent the same knot?
Sam Nelson The Algebra of Knots
Reidemeister Moves
Sam Nelson The Algebra of Knots
Example
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
Knot Invariants
Examples:
Alexander/Jones/HOMFLYpt/Kauffman polynomials
Knot group
Hyperbolic Volume
TQFTs
Khovanov Homology
Sam Nelson The Algebra of Knots
Knot Invariants
Examples:
Alexander/Jones/HOMFLYpt/Kauffman polynomials
Knot group
Hyperbolic Volume
TQFTs
Khovanov Homology
Sam Nelson The Algebra of Knots
Knot Invariants
Examples:
Alexander/Jones/HOMFLYpt/Kauffman polynomials
Knot group
Hyperbolic Volume
TQFTs
Khovanov Homology
Sam Nelson The Algebra of Knots
Knot Invariants
Examples:
Alexander/Jones/HOMFLYpt/Kauffman polynomials
Knot group
Hyperbolic Volume
TQFTs
Khovanov Homology
Sam Nelson The Algebra of Knots
Knot Invariants
Examples:
Alexander/Jones/HOMFLYpt/Kauffman polynomials
Knot group
Hyperbolic Volume
TQFTs
Khovanov Homology
Sam Nelson The Algebra of Knots
Knot Invariants
Examples:
Alexander/Jones/HOMFLYpt/Kauffman polynomials
Knot group
Hyperbolic Volume
TQFTs
Khovanov Homology
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
Kei
(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
Sam Nelson The Algebra of Knots
Kei
(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
Sam Nelson The Algebra of Knots
Kei
(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
Sam Nelson The Algebra of Knots
Kei
(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
Sam Nelson The Algebra of Knots
Kei Axioms
x . x = x
Sam Nelson The Algebra of Knots
Kei Axioms
(x . y) . y = x
Sam Nelson The Algebra of Knots
Kei Axioms
(x . y) . z = (x . z) . (y . z)
Sam Nelson The Algebra of Knots
Kei
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)
Sam Nelson The Algebra of Knots
Kei
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)
Sam Nelson The Algebra of Knots
Kei
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)
Sam Nelson The Algebra of Knots
Kei
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)
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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.
Sam Nelson The Algebra of Knots
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.
Sam Nelson The Algebra of Knots
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.
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
The counting invariant
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
Sam Nelson The Algebra of Knots
The counting invariant
Because of the kei axioms, every valid labeling of a diagrambefore a move corresponds to a unique valid labeling after amove.
Sam Nelson The Algebra of Knots
The counting invariant
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!
Sam Nelson The Algebra of Knots
The counting invariant
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!
Sam Nelson The Algebra of Knots
The counting invariant
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
Sam Nelson The Algebra of Knots
The counting invariant
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
Sam Nelson The Algebra of Knots
The counting invariant
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
Sam Nelson The Algebra of Knots
The counting invariant
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
Sam Nelson The Algebra of Knots
The counting invariant
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
Sam Nelson The Algebra of Knots
The counting invariant
The unknot has three labelings by our three-element kei:
Sam Nelson The Algebra of Knots
A knot invariant
The trefoil has nine labelings by our three-element kei:
Sam Nelson The Algebra of Knots
A knot invariant
3 6= 9, and the kei counting invariant shows that no sequence ofReidemeister moves can unknot the trefoil.
Sam Nelson The Algebra of Knots
Generalizations of Kei
An oriented knot has a preferred direction of travel.
Sam Nelson The Algebra of Knots
Generalizations of Kei
Orienting K changes axiom (ii) to allow .−1 to be differentfrom .; the resulting object is called a quandle
Sam Nelson The Algebra of Knots
Generalizations of Kei
Orienting K changes axiom (ii) to allow .−1 to be differentfrom .; the resulting object is called a quandle
Sam Nelson The Algebra of Knots
Generalizations of Kei
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
Sam Nelson The Algebra of Knots
Generalizations of Kei
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
Sam Nelson The Algebra of Knots
Generalizations of Kei
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
Sam Nelson The Algebra of Knots
Generalizations of Kei
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
Sam Nelson The Algebra of Knots
Generalizations of Kei
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
Sam Nelson The Algebra of Knots
Generalizations of Kei
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
Sam Nelson The Algebra of Knots
Generalizations of Kei
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
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
Sam Nelson The Algebra of Knots
Enhanced Invariants
Example: For each labeling, count uc where c is the number oflabels used. This is called the image-enhanced countinginvariant, denoted ΦIm
X .
Sam Nelson The Algebra of Knots
Enhanced Invariants
ΦImX (Unknot) = u + u + u = 3u
Sam Nelson The Algebra of Knots
Enhanced Invariants
ΦImX (31) = 3u + 6u3
Sam Nelson The Algebra of Knots
Enhancements
Enhancements can come from the general structure of thelabeling object.
Image enhancements
Writhe enhancements
Quandle polynomials
Column groups
Sam Nelson The Algebra of Knots
Enhancements
Enhancements can come from the general structure of thelabeling object.
Image enhancements
Writhe enhancements
Quandle polynomials
Column groups
Sam Nelson The Algebra of Knots
Enhancements
Enhancements can come from the general structure of thelabeling object.
Image enhancements
Writhe enhancements
Quandle polynomials
Column groups
Sam Nelson The Algebra of Knots
Enhancements
Enhancements can come from the general structure of thelabeling object.
Image enhancements
Writhe enhancements
Quandle polynomials
Column groups
Sam Nelson The Algebra of Knots
Enhancements
Enhancements can come from the general structure of thelabeling object.
Image enhancements
Writhe enhancements
Quandle polynomials
Column groups
Sam Nelson The Algebra of Knots
Enhancements
Enhancements can come from using special types of labelingobjects.
Symplectic quandle invariants
Bilinear biquandle invariants
Coxeter rack invariants
(t, s)-rack invariants
Sam Nelson The Algebra of Knots
Enhancements
Enhancements can come from using special types of labelingobjects.
Symplectic quandle invariants
Bilinear biquandle invariants
Coxeter rack invariants
(t, s)-rack invariants
Sam Nelson The Algebra of Knots
Enhancements
Enhancements can come from using special types of labelingobjects.
Symplectic quandle invariants
Bilinear biquandle invariants
Coxeter rack invariants
(t, s)-rack invariants
Sam Nelson The Algebra of Knots
Enhancements
Enhancements can come from using special types of labelingobjects.
Symplectic quandle invariants
Bilinear biquandle invariants
Coxeter rack invariants
(t, s)-rack invariants
Sam Nelson The Algebra of Knots
Enhancements
Enhancements can come from using special types of labelingobjects.
Symplectic quandle invariants
Bilinear biquandle invariants
Coxeter rack invariants
(t, s)-rack invariants
Sam Nelson The Algebra of Knots
Enhancements
Enhancements can also come from additional structures.
Quandle 2-cocycles
Rack shadows
Rack algebras
Sam Nelson The Algebra of Knots
Enhancements
Enhancements can also come from additional structures.
Quandle 2-cocycles
Rack shadows
Rack algebras
Sam Nelson The Algebra of Knots
Enhancements
Enhancements can also come from additional structures.
Quandle 2-cocycles
Rack shadows
Rack algebras
Sam Nelson The Algebra of Knots
Enhancements
Enhancements can also come from additional structures.
Quandle 2-cocycles
Rack shadows
Rack algebras
Sam Nelson The Algebra of Knots
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!
Sam Nelson The Algebra of Knots
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!
Sam Nelson The Algebra of Knots
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!
Sam Nelson The Algebra of Knots
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!
Sam Nelson The Algebra of Knots
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!
Sam Nelson The Algebra of Knots
Thanks for Listening!
Sam Nelson The Algebra of Knots
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