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The Algebra of Knots 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 .
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