The Algebra of Knots
Sam Nelson
Claremont McKenna College
Sam Nelson The Algebra of Knots Addition comes from unions:
Familiar Operations and Sets
Sam Nelson The Algebra of Knots Familiar Operations and Sets
Addition comes from unions:
Sam Nelson The Algebra of Knots Multiplication comes from matched pairs:
Familiar Operations and Sets
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 of the 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 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?
Sam Nelson The Algebra of Knots 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?
Many useful real-world quantities behave like sets
Sam Nelson The Algebra of Knots However, not every quantity behaves so simply Examples: waves interfere, particles become entangled Let us now consider operations inspired by knots:
Why sets?
Many useful real-world quantities behave like sets Examples: lengths, masses, volumes, money
Sam Nelson The Algebra of Knots Examples: waves interfere, particles become entangled Let us now consider operations inspired by knots:
Why sets?
Many useful real-world quantities behave like sets Examples: lengths, masses, volumes, money However, not every quantity behaves so simply
Sam Nelson The Algebra of Knots Let us now consider operations inspired by 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
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 simple - does not intersect itself closed - has no loose endpoints, i.e. forms a loop
Knots
Definition: A knot is a simple closed curve in three-dimensional space.
Sam Nelson The Algebra of Knots closed - has no loose endpoints, i.e. forms a loop
Knots
Definition: A knot is a simple closed curve in three-dimensional space. simple - does not intersect itself
Sam Nelson The Algebra of Knots Knots
Definition: A knot is a simple closed curve in three-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 in space will result in different diagrams of the same knot.
Sam Nelson The Algebra of Knots moving the knot in space stretching or shrinking in a continuous way but not cutting and retying
Topology
Knots are topological objects, meaning that we consider two knots the same if one can be changed into the other by:
Sam Nelson The Algebra of Knots stretching or shrinking in a continuous way but not cutting and retying
Topology
Knots are topological objects, meaning that we consider two knots the same if one can be changed into the other by: moving the knot in space
Sam Nelson The Algebra of Knots but not cutting and retying
Topology
Knots are topological objects, meaning that we consider two knots the same if one can be changed into the other by: moving the knot in space stretching or shrinking in a continuous way
Sam Nelson The Algebra of Knots Topology
Knots are topological objects, meaning that we consider two knots 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 Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined in part by how they’re knotted Certain antibiotics work by blocking the action of molecules called topoisomerase which change how DNA is knotted; blocking the unknotting of the DNA stops the bacteria reproducing Perhaps surprisingly, the mathematics of knots is relevant to the search for a theory of quantum gravity, a major unsolved problem in physics Besides, knots are just fun!
Why Knots?
Sam Nelson The Algebra of Knots Certain antibiotics work by blocking the action of molecules called topoisomerase which change how DNA is knotted; blocking the unknotting of the DNA stops the bacteria reproducing Perhaps surprisingly, the mathematics of knots is relevant to the search for a theory of quantum gravity, a major unsolved problem in physics Besides, knots are just fun!
Why Knots?
Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined in part by how they’re knotted
Sam Nelson The Algebra of Knots Perhaps surprisingly, the mathematics of knots is relevant to the search for a theory of quantum gravity, a major unsolved problem in physics Besides, knots are just fun!
Why Knots?
Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined in part by how they’re knotted Certain antibiotics work by blocking the action of molecules called topoisomerase which change how DNA is knotted; blocking the unknotting of the DNA stops the bacteria reproducing
Sam Nelson The Algebra of Knots Besides, knots are just fun!
Why Knots?
Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined in part by how they’re knotted Certain antibiotics work by blocking the action of molecules called topoisomerase which change how DNA is knotted; blocking the unknotting of the DNA stops the bacteria reproducing Perhaps surprisingly, the mathematics of knots is relevant to the search for a theory of quantum gravity, a major unsolved problem in physics
Sam Nelson The Algebra of Knots Why Knots?
Many molecules (polymers, protein, DNA) are knots, and their chemical properties are determined in part by how they’re knotted Certain antibiotics work by blocking the action of molecules called topoisomerase which change how DNA is knotted; blocking the unknotting of the DNA stops the bacteria reproducing Perhaps surprisingly, the mathematics of knots is relevant to the search for a theory of quantum gravity, a major unsolved problem in physics Besides, knots are just fun!
Sam Nelson The Algebra of Knots The main problem
Given two knot diagrams K and K0, how can we tell whether they 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 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 K0 have different invariant values, they represent different knots
Knot Invariants
Sam Nelson The Algebra of Knots Get the same value for all diagrams of the same knot Unchanged by Reidemeister moves If K and K0 have different invariant values, they represent different knots
Knot Invariants
Quantities we can compute from a knot diagram
Sam Nelson The Algebra of Knots Unchanged by Reidemeister moves If K and K0 have different invariant values, they represent different knots
Knot Invariants
Quantities we can compute from a knot diagram Get the same value for all diagrams of the same knot
Sam Nelson The Algebra of Knots If K and K0 have different invariant values, they represent different 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
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 K0 have different invariant values, they represent different knots
Sam Nelson The Algebra of Knots Alexander/Jones/HOMFLYpt/Kauffman polynomials Knot group Hyperbolic Volume TQFTs Khovanov Homology
Knot Invariants
Examples:
Sam Nelson The Algebra of Knots Knot group Hyperbolic Volume TQFTs Khovanov Homology
Knot Invariants
Examples: Alexander/Jones/HOMFLYpt/Kauffman polynomials
Sam Nelson The Algebra of Knots Hyperbolic Volume TQFTs Khovanov Homology
Knot Invariants
Examples: Alexander/Jones/HOMFLYpt/Kauffman polynomials Knot group
Sam Nelson The Algebra of Knots TQFTs Khovanov Homology
Knot Invariants
Examples: Alexander/Jones/HOMFLYpt/Kauffman polynomials Knot group Hyperbolic Volume
Sam Nelson The Algebra of Knots Khovanov Homology
Knot Invariants
Examples: Alexander/Jones/HOMFLYpt/Kauffman polynomials Knot group Hyperbolic Volume TQFTs
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 Define an algebraic operation . from knot diagrams Use Reidemeister moves to determine properties of the operation Find operations satisfying these axioms either by combining existing operations or by making operation tables
An algebraic knot invariant
Sam Nelson The Algebra of Knots Use Reidemeister moves to determine properties of the operation Find operations satisfying these axioms either by combining existing operations or by making operation tables
An algebraic knot invariant
Define an algebraic operation . from knot diagrams
Sam Nelson The Algebra of Knots Find operations satisfying these axioms either by combining existing operations or by making operation tables
An algebraic knot invariant
Define an algebraic operation . from knot diagrams Use Reidemeister moves to determine properties of the operation
Sam Nelson The Algebra of Knots An algebraic knot invariant
Define an algebraic operation . from knot diagrams Use Reidemeister moves to determine properties of the operation Find operations satisfying these axioms either by combining existing operations or by making operation tables
Sam Nelson The Algebra of Knots Attach a label to each arc in a knot diagram When x goes under y, the result is x . y
Kei (a.k.a. Involutory Quandles)
Sam Nelson The Algebra of Knots When x goes under y, the result is x . y
Kei (a.k.a. Involutory Quandles)
Attach a label to each arc in a knot diagram
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 (i) x . x = x (ii) (x . y) . y = x (iii) (x . y) . z = (x . z) . (y . z)
Kei
Thus, a kei is a set of labels with an operation . such that for all x, y, z we have
Sam Nelson The Algebra of Knots (ii) (x . y) . y = x (iii) (x . y) . z = (x . z) . (y . z)
Kei
Thus, a kei is a set of labels with an operation . such that for all x, y, z we have (i) x . x = x
Sam Nelson The Algebra of Knots (iii) (x . y) . z = (x . z) . (y . z)
Kei
Thus, a kei is a set of labels with an operation . such that for all x, y, z we have (i) x . x = x (ii) (x . y) . y = x
Sam Nelson The Algebra of Knots Kei
Thus, a kei is a set of labels with an operation . such that for all 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 We can define kei operations using existing operations. For example, 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 it satisfies the axioms. For example,
x . x = 2x − x = x
Kei Example
Sam Nelson The Algebra of Knots To see that this is a kei operation, we need to verify that it satisfies the axioms. For example,
x . x = 2x − x = x
Kei Example
We can define kei operations using existing operations. For example, the integers Z have a kei operation given by
x . y = 2y − x.
Sam Nelson The Algebra of Knots Kei Example
We can define kei operations using existing operations. For example, 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 it satisfies the axioms. For example,
x . x = 2x − x = x
Sam Nelson The Algebra of Knots We can also specify a kei operation with an operation table, just like a multiplication table:
. 1 2 3 1 1 3 2 2 3 2 1 3 2 1 3
Checking the axioms here must be done case-by-case and is best handled by computer code.
Kei Example
Sam Nelson The Algebra of Knots Checking the axioms here must be done case-by-case and is best handled by computer code.
Kei Example
We can also specify a kei operation with an operation table, just like a multiplication table:
. 1 2 3 1 1 3 2 2 3 2 1 3 2 1 3
Sam Nelson The Algebra of Knots Kei Example
We can also specify a kei operation with an operation table, just like a multiplication table:
. 1 2 3 1 1 3 2 2 3 2 1 3 2 1 3
Checking the axioms here must be done case-by-case and is best 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 the crossing condition at every crossing.
. 1 2 3 1 1 3 2 2 3 2 1 3 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 the crossing condition at any crossing, it is invalid.
. 1 2 3 1 1 3 2 2 3 2 1 3 2 1 3
Sam Nelson The Algebra of Knots The counting invariant
Because of the kei axioms, every valid labeling of a diagram before a move corresponds to a unique valid labeling after a move.
Sam Nelson The Algebra of Knots we count the valid labelings!
The counting invariant
Any two diagrams of the same knot will have same number of valid labelings by your favorite kei. So, to tell knots apart,
Sam Nelson The Algebra of Knots The counting invariant
Any two diagrams of the same knot will have same number of valid labelings by your favorite kei. So, to tell knots apart,
we count the valid labelings!
Sam Nelson The Algebra of Knots If our set of labels is finite, there are only finitely many possible labelings So we can list all possible labelings and count how many are valid This number will be the same for all diagrams of K So it is a knot invariant, called the kei counting invariant
The counting invariant
Sam Nelson The Algebra of Knots So we can list all possible labelings and count how many are valid This number will be the same for all diagrams of K So it is a knot invariant, called the kei counting invariant
The counting invariant
If our set of labels is finite, there are only finitely many possible labelings
Sam Nelson The Algebra of Knots This number will be the same for all diagrams of K So it is a knot invariant, called the kei counting invariant
The counting invariant
If our set of labels is finite, there are only finitely many possible labelings So we can list all possible labelings and count how many are valid
Sam Nelson The Algebra of Knots So it is a knot invariant, called the kei counting invariant
The counting invariant
If our set of labels is finite, there are only finitely many possible labelings So we can list all possible labelings and count how many are valid This number will be the same for all diagrams of K
Sam Nelson The Algebra of Knots The counting invariant
If our set of labels is finite, there are only finitely many possible labelings So we can list all possible labelings and count how many are 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 of Reidemeister 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 Orienting K changes axiom (ii) to allow .−1 to be different from .; the resulting object is called a quandle
Generalizations of Kei
Sam Nelson The Algebra of Knots Generalizations of Kei
Orienting K changes axiom (ii) to allow .−1 to be different from .; the resulting object is called a quandle
Sam Nelson The Algebra of Knots Replacing the type I move with a doubled version gives us framed knots which are like physical knots This eliminates quandle axiom (i); the resulting object is called a rack
Generalizations of Kei
Sam Nelson The Algebra of Knots This eliminates quandle axiom (i); the resulting object is called a rack
Generalizations of Kei
Replacing the type I move with a doubled version gives us framed knots which are like physical knots
Sam Nelson The Algebra of Knots Generalizations of Kei
Replacing the type I move with a doubled version gives us framed knots which are like physical knots This eliminates quandle axiom (i); the resulting object is called a rack
Sam Nelson The Algebra of Knots Dividing the arcs at over-crossings to get semi-arcs lets both inputs act on each other The resulting algebraic structures are called bikei, biquandles and biracks These are solutions to the Yang-Baxter Equation from statistical mechanics
Generalizations of Kei
Sam Nelson The Algebra of Knots The resulting algebraic structures are called bikei, biquandles and biracks These are solutions to the Yang-Baxter Equation from statistical mechanics
Generalizations of Kei
Dividing the arcs at over-crossings to get semi-arcs lets both inputs act on each other
Sam Nelson The Algebra of Knots These are solutions to the Yang-Baxter Equation from statistical mechanics
Generalizations of Kei
Dividing the arcs at over-crossings to get semi-arcs lets both inputs act on each other The resulting algebraic structures are called bikei, biquandles and biracks
Sam Nelson The Algebra of Knots Generalizations of Kei
Dividing the arcs at over-crossings to get semi-arcs lets both inputs act on each other The resulting algebraic structures are called bikei, biquandles and biracks These are solutions to the Yang-Baxter Equation from statistical mechanics
Sam Nelson The Algebra of Knots A signature is a quantity computable from a labeled knot diagram which is invariant under labeled moves Counting signatures instead of labelings defines stronger invariants
Enhanced Invariants
Sam Nelson The Algebra of Knots Counting signatures instead of labelings defines stronger invariants
Enhanced Invariants
A signature is a quantity computable from a labeled knot diagram which is invariant under labeled moves
Sam Nelson The Algebra of Knots Enhanced Invariants
A signature is a quantity computable from a labeled knot diagram which is invariant under labeled moves Counting signatures instead of labelings defines stronger invariants
Sam Nelson The Algebra of Knots Enhanced Invariants
Example: For each labeling, count uc where c is the number of labels used. This is called the image-enhanced counting Im invariant, denoted ΦX .
Sam Nelson The Algebra of Knots Enhanced Invariants
Im ΦX (Unknot) = u + u + u = 3u
Sam Nelson The Algebra of Knots Enhanced Invariants
Im 3 ΦX (31) = 3u + 6u
Sam Nelson The Algebra of Knots Image enhancements Writhe enhancements Quandle polynomials Column groups
Enhancements
Enhancements can come from the general structure of the labeling object.
Sam Nelson The Algebra of Knots Writhe enhancements Quandle polynomials Column groups
Enhancements
Enhancements can come from the general structure of the labeling object. Image enhancements
Sam Nelson The Algebra of Knots Quandle polynomials Column groups
Enhancements
Enhancements can come from the general structure of the labeling object. Image enhancements Writhe enhancements
Sam Nelson The Algebra of Knots Column groups
Enhancements
Enhancements can come from the general structure of the labeling object. Image enhancements Writhe enhancements Quandle polynomials
Sam Nelson The Algebra of Knots Enhancements
Enhancements can come from the general structure of the labeling object. Image enhancements Writhe enhancements Quandle polynomials Column groups
Sam Nelson The Algebra of Knots Symplectic quandle invariants Bilinear biquandle invariants Coxeter rack invariants (t, s)-rack invariants
Enhancements
Enhancements can come from using special types of labeling objects.
Sam Nelson The Algebra of Knots Bilinear biquandle invariants Coxeter rack invariants (t, s)-rack invariants
Enhancements
Enhancements can come from using special types of labeling objects. Symplectic quandle invariants
Sam Nelson The Algebra of Knots Coxeter rack invariants (t, s)-rack invariants
Enhancements
Enhancements can come from using special types of labeling objects. Symplectic quandle invariants Bilinear biquandle invariants
Sam Nelson The Algebra of Knots (t, s)-rack invariants
Enhancements
Enhancements can come from using special types of labeling objects. Symplectic quandle invariants Bilinear biquandle invariants Coxeter rack invariants
Sam Nelson The Algebra of Knots Enhancements
Enhancements can come from using special types of labeling objects. Symplectic quandle invariants Bilinear biquandle invariants Coxeter rack invariants (t, s)-rack invariants
Sam Nelson The Algebra of Knots Quandle 2-cocycles Rack shadows Rack algebras
Enhancements
Enhancements can also come from additional structures.
Sam Nelson The Algebra of Knots Rack shadows Rack algebras
Enhancements
Enhancements can also come from additional structures. Quandle 2-cocycles
Sam Nelson The Algebra of Knots Rack algebras
Enhancements
Enhancements can also come from additional structures. Quandle 2-cocycles Rack shadows
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 Algebra from knots seems weird at first, but the algebraic structures arising from knots have application in biology, chemistry, physics... and who knows where else? You can help find out!
Tying it all up...
Sam Nelson The Algebra of Knots but the algebraic structures arising from knots have application in biology, chemistry, physics... and who knows where else? You can help find out!
Tying it all up...
Algebra from knots seems weird at first,
Sam Nelson The Algebra of Knots and who knows where else? You can help find out!
Tying it all up...
Algebra from knots seems weird at first, but the algebraic structures arising from knots have application in biology, chemistry, physics...
Sam Nelson The Algebra of Knots You can help find out!
Tying it all up...
Algebra from knots seems weird at first, but the algebraic structures arising from knots have application in biology, chemistry, physics... and who knows where else?
Sam Nelson The Algebra of Knots Tying it all up...
Algebra from knots seems weird at first, but the algebraic structures arising from knots have application 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