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Knot Theory Talk

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Knot Theory Talk A brief Incursion into Knot Theory Eduardo Balreira Trinity University Mathematics Department Major Seminar, Fall 2008 (Balreira - Trinity University) Knot Theory Major Seminar 1 / 31 Outline 1 A Fundamental Problem (Balreira - Trinity University) Knot Theory Major Seminar 2 / 31 Outline 1 A Fundamental Problem 2 Knot Theory (Balreira - Trinity University) Knot Theory Major Seminar 2 / 31 Outline 1 A Fundamental Problem 2 Knot Theory 3 Reidemeister Moves (Balreira - Trinity University) Knot Theory Major Seminar 2 / 31 Outline 1 A Fundamental Problem 2 Knot Theory 3 Reidemeister Moves 4 Invariants Colorability The Knot Group (Balreira - Trinity University) Knot Theory Major Seminar 2 / 31 A Fundamental Problem Given two objects, how to tell them apart? When are two surfaces homeomorphic? ? (Balreira - Trinity University) Knot Theory Major Seminar 3 / 31 A Fundamental Problem Given two objects, how to tell them apart? When are two surfaces homeomorphic? ? When are two knots equivalent? ? (Balreira - Trinity University) Knot Theory Major Seminar 3 / 31 A Fundamental Problem Main Research Area Injectivity via Geometric and Topological Methods Foliation Theory (Balreira - Trinity University) Knot Theory Major Seminar 4 / 31 A Fundamental Problem Main Research Area Injectivity via Geometric and Topological Methods Foliation Theory Spectral Theory (Balreira - Trinity University) Knot Theory Major Seminar 4 / 31 A Fundamental Problem Main Research Area Injectivity via Geometric and Topological Methods Foliation Theory Spectral Theory Variational Calculus (Balreira - Trinity University) Knot Theory Major Seminar 4 / 31 A Fundamental Problem Main Research Area Injectivity via Geometric and Topological Methods Foliation Theory Spectral Theory Variational Calculus Global Embedding of Submanifolds (Balreira - Trinity University) Knot Theory Major Seminar 4 / 31 Sample Topics (Balreira - Trinity University) Knot Theory Major Seminar 5 / 31 Sample Topics Project Problem (Foliation Theory) Discuss Fibrations versus Foliations, e.g., characterize the foliations of the Euclidean Plane, that is, R2. (Balreira - Trinity University) Knot Theory Major Seminar 5 / 31 Sample Topics Project Problem (Foliation Theory) Discuss Fibrations versus Foliations, e.g., characterize the foliations of the Euclidean Plane, that is, R2. Project Problem (Spectral Theory) Discuss the eigenvalues of the Laplacian on a bounded region of Rn, e.g., Rayleigh and Min-Max Methods. (Balreira - Trinity University) Knot Theory Major Seminar 5 / 31 Sample Topics (Balreira - Trinity University) Knot Theory Major Seminar 6 / 31 Sample Topics Project Problem (Variational Calculus) Discuss any result in the Geometric Analysis report, e.g., understand the MPT and its proof. (Balreira - Trinity University) Knot Theory Major Seminar 6 / 31 Sample Topics Project Problem (Variational Calculus) Discuss any result in the Geometric Analysis report, e.g., understand the MPT and its proof. Project Problem (Geometry) Learn about Classification of Surfaces via the Euler Characteristic and/or understand the Gauss-Bonnet formula K dA = 2πχ(M) ZM (Balreira - Trinity University) Knot Theory Major Seminar 6 / 31 Knot Theory When can tell the difference between Knots? Are the knots below the same? (Balreira - Trinity University) Knot Theory Major Seminar 7 / 31 Knot Theory When can tell the difference between Knots? How about these? (Balreira - Trinity University) Knot Theory Major Seminar 8 / 31 Knot Theory Some Formalism A knot is an injective map h : S1 → R3 Picture in the plane (or slide) - Diagram with crossing (Balreira - Trinity University) Knot Theory Major Seminar 9 / 31 Knot Theory Some Formalism A knot is an injective map h : S1 → R3 Picture in the plane (or slide) - Diagram with crossing Tame Knots (Balreira - Trinity University) Knot Theory Major Seminar 9 / 31 Knot Theory Some Formalism A knot is an injective map h : S1 → R3 Picture in the plane (or slide) - Diagram with crossing Tame Knots ◮ Finite Number of arcs (Balreira - Trinity University) Knot Theory Major Seminar 9 / 31 Knot Theory Some Formalism A knot is an injective map h : S1 → R3 Picture in the plane (or slide) - Diagram with crossing Tame Knots ◮ Finite Number of arcs ◮ Only two strands at a crossing (Balreira - Trinity University) Knot Theory Major Seminar 9 / 31 Knot Theory Some Formalism A knot is an injective map h : S1 → R3 Picture in the plane (or slide) - Diagram with crossing Tame Knots ◮ Finite Number of arcs ◮ Only two strands at a crossing ◮ “nice” (Balreira - Trinity University) Knot Theory Major Seminar 9 / 31 Knot Theory Some Formalism A knot is an injective map h : S1 → R3 Picture in the plane (or slide) - Diagram with crossing Tame Knots ◮ Finite Number of arcs ◮ Only two strands at a crossing ◮ “nice” Invariant Property: K1 ∼ K2 if, (Balreira - Trinity University) Knot Theory Major Seminar 9 / 31 Knot Theory Some Formalism A knot is an injective map h : S1 → R3 Picture in the plane (or slide) - Diagram with crossing Tame Knots ◮ Finite Number of arcs ◮ Only two strands at a crossing ◮ “nice” Invariant Property: K1 ∼ K2 if, There is a homeomorphism ϕ : R3 → R3 such that ϕ(K1)= K2 (Balreira - Trinity University) Knot Theory Major Seminar 9 / 31 Knot Theory Isotopy Problem The map ϕ itself must be “nice” ←→ Isotopic to the Identity No situations such as: (Balreira - Trinity University) Knot Theory Major Seminar 10 / 31 Knot Theory - Diagrams Two Knots with similar Diagram must be the same. Trefoil Braided Trefoil (Balreira - Trinity University) Knot Theory Major Seminar 11 / 31 Knot Theory - Diagrams Two Knots with similar Diagram must be the same. Trefoil Braided Trefoil (Balreira - Trinity University) Knot Theory Major Seminar 11 / 31 Knot Theory - Diagrams Two Knots with similar Diagram must be the same. Trefoil Braided Trefoil (Balreira - Trinity University) Knot Theory Major Seminar 11 / 31 Knot Theory - Diagrams Two Knots with similar Diagram must be the same. Trefoil Braided Trefoil (Balreira - Trinity University) Knot Theory Major Seminar 11 / 31 Knot Theory - Diagrams Two Knots with similar Diagram must be the same. Trefoil Braided Trefoil (Balreira - Trinity University) Knot Theory Major Seminar 11 / 31 Knot Theory - Diagrams Even if not exactly the diagram same, they could be the same... (Balreira - Trinity University) Knot Theory Major Seminar 12 / 31 Moves that preserve the Knot Reidemeister Moves Type I: Put or Take out a kink. (Balreira - Trinity University) Knot Theory Major Seminar 13 / 31 Moves that preserve the Knot Reidemeister Moves Type I: Put or Take out a kink. Type II: Slide a strand over/under to creat/remove two crossings. (Balreira - Trinity University) Knot Theory Major Seminar 13 / 31 Moves that preserve the Knot Reidemeister Moves Type I: Put or Take out a kink. Type II: Slide a strand over/under to creat/remove two crossings. Type III: Slide a strand across a crossing. (Balreira - Trinity University) Knot Theory Major Seminar 13 / 31 Type I Put or Take out a kink. (Balreira - Trinity University) Knot Theory Major Seminar 14 / 31 Type II Slide a strand over/under to creat/remove two crossings. (Balreira - Trinity University) Knot Theory Major Seminar 15 / 31 Type III Slide a strand across a crossing (Balreira - Trinity University) Knot Theory Major Seminar 16 / 31 Unknotting a Knot Using Reidemeister moves (Balreira - Trinity University) Knot Theory Major Seminar 17 / 31 Unknotting a Knot Using Reidemeister moves (Balreira - Trinity University) Knot Theory Major Seminar 17 / 31 Unknotting a Knot Using Reidemeister moves (Balreira - Trinity University) Knot Theory Major Seminar 17 / 31 Unknotting a Knot Using Reidemeister moves (Balreira - Trinity University) Knot Theory Major Seminar 17 / 31 Knot Theory A Possible classification? (Balreira - Trinity University) Knot Theory Major Seminar 18 / 31 Knot Theory A Possible classification? Theorem (Reidemeister, 1932) Two Knot diagrams of the same knot can be deformed into each other by a finite number of moves of Type I, II, and III. (Balreira - Trinity University) Knot Theory Major Seminar 18 / 31 Knot Theory A Possible classification? Theorem (Reidemeister, 1932) Two Knot diagrams of the same knot can be deformed into each other by a finite number of moves of Type I, II, and III. Project Problem Discuss the result above, its proof, and its significance in other areas. For instance, Physics - the structure of the atom and Chemistry - the structure of the DNA replication. (Balreira - Trinity University) Knot Theory Major Seminar 18 / 31 Invariants Colorability (Balreira - Trinity University) Knot Theory Major Seminar 19 / 31 Invariants Colorability Unknotting number (Balreira - Trinity University) Knot Theory Major Seminar 19 / 31 Invariants Colorability Unknotting number Genus of a knot (Balreira - Trinity University) Knot Theory Major Seminar 19 / 31 Invariants Colorability Unknotting number Genus of a knot Knot Group (Balreira - Trinity University) Knot Theory Major Seminar 19 / 31 Invariants Colorability Unknotting number Genus of a knot Knot Group Polynomial Invariants (Balreira - Trinity University) Knot Theory Major Seminar 19 / 31 Sample Problems in Knot Theory (Balreira - Trinity University) Knot Theory Major Seminar 20 / 31 Sample Problems in Knot Theory Project Problem (Colorability) Understand the definition and application as well as generalizations for mod p colorability. This project has a Topological and Number Theoretical flavor. (Balreira - Trinity University) Knot Theory Major Seminar 20 / 31 Sample Problems in Knot
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