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An Introduction to Knot Theory

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An Introduction to Knot Theory Introduction Knot diagrams An introduction to Knot Theory Ioannis Diamantis China Agricultural University, International College Beijing Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History WELCOME! Usefull Information Dr. Ioannis Diamantis (Yannis), BSc, MSc, PhD in Mathematics Email: [email protected] or [email protected] Office: 313 ICB Website: http://yannismath.webnode.gr/ Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History Title of Thesis; Athens 2015 The HOMFLYPT skein module of the lens spaces L(p; 1) via braids PhD. Examination Committee Prof. S. Lambropoulou (Nat.Tech.U.Athens); Thesis Advisor Prof. Jozef Przytycki (George Washinghton U.) Prof. Panos Papazoglu (Oxford University) Prof. Cameron Gordon (U. of Austin Texas) Prof. Them. Rassias (N.T.U.A.) Prof. Epam. Kechagias (U. of Ioannina) Lecturer Dim. Kodokostas (N.T.U.A.) Prof. Louis Kauffman (U.I.Chicago); Unofficial Examiner Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History Research Program of Excellency \Thalis" \Algebraic modeling of topological and computational structures and applications" Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History Ioannis Diamantis An introduction to Knot Theory To make a study of knottedness, the knotted part of the string must be trapped. One way to do this is to imagine an infinitely long string which is a straight line outside the region containing 3 3 the knot (compactification of R ∼ S ). Figure 1: Forming a knot from a piece of string Introduction Set up Knot diagrams History We are all familiar with everyday knots - the kind that we use to tie up parcels, shoelaces, and so on. These knots can be untied, and retied in the same or different ways. By manipulating the string we can let the 'knots' escape. Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History We are all familiar with everyday knots - the kind that we use to tie up parcels, shoelaces, and so on. These knots can be untied, and retied in the same or different ways. By manipulating the string we can let the 'knots' escape. To make a study of knottedness, the knotted part of the string must be trapped. One way to do this is to imagine an infinitely long string which is a straight line outside the region containing 3 3 the knot (compactification of R ∼ S ). Figure 1: Forming a knot from a piece of string Ioannis Diamantis An introduction to Knot Theory Knot theory is the study of the topological properties of loops embedded in 3-dimensional space. Mathematical knots are modeled on the physical variety, and we allow a knot to be deformed as if it were made of a thin, flexible, elastic thread. We shall begin by making these ideas more precise. Introduction Set up Knot diagrams History A simpler way is to join the ends to form a loop. Figure 2: Forming a knot from a piece of string Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History A simpler way is to join the ends to form a loop. Figure 2: Forming a knot from a piece of string Knot theory is the study of the topological properties of loops embedded in 3-dimensional space. Mathematical knots are modeled on the physical variety, and we allow a knot to be deformed as if it were made of a thin, flexible, elastic thread. We shall begin by making these ideas more precise. Ioannis Diamantis An introduction to Knot Theory Definition 3 Let A; B ⊂ R . A function f : A ! B is called a homeomorphism if it's 1-1, continuous and has a continuous inverse (f −1). The sets A; B are then called homeomorphic. Introduction Set up Knot diagrams History Definition 3 A knot K ⊂ R is a subset of points homeomorphic to a circle. Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History Definition 3 A knot K ⊂ R is a subset of points homeomorphic to a circle. Definition 3 Let A; B ⊂ R . A function f : A ! B is called a homeomorphism if it's 1-1, continuous and has a continuous inverse (f −1). The sets A; B are then called homeomorphic. Ioannis Diamantis An introduction to Knot Theory Definition A knot is an embedding of S1 into S3, while a link is an embedding of N-copies of S1 to S3. Figure 3: (a) The unknot, (b) The trefoil knot Introduction Set up Knot diagrams History Definition 3 Let G ⊂ R . An embedding of G is the image of G under a homeomorphism f . Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History Definition 3 Let G ⊂ R . An embedding of G is the image of G under a homeomorphism f . Definition A knot is an embedding of S1 into S3, while a link is an embedding of N-copies of S1 to S3. Figure 3: (a) The unknot, (b) The trefoil knot Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History Figure 4: A Wild knot covering the Cantor's continuum (from "Knots" by A. Sossinsky). Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History Figure 5: Knots and Links Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History We now require mathematical knots to be flexible. We need to model the physical manipulation by which a thread can be deformed into different positions. Figure 6: Deforming a knot doesn't change it Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History Figure 7: A nonstandard picture of the trefoil knot Figure 8: The unknot Main Problem How can we distinguish one knot from another? Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History If you have learned any algebraic topology, you will be familiar with the deformation known as homotopy. Definition 3 A homotopy of a space X ⊂ R is a continuous map 3 h : X × [0; 1] ! R . 3 The restriction of h to level t 2 [0; 1] is ht : X × t ! R . Here, t is chosen deliberately to indicate time, and the images ht (X ) for 3 increasing values of t show the evolution of X in R . We require that h0 is the identity map. All the ht 's inherit the continuity property of h. Ioannis Diamantis An introduction to Knot Theory Unfortunately, isotopy is no use for deforming knots either. Even though a knot cannot pass through itself under isotopy, all knots are isotopic to the trivial knot. To see why this is so, imagine pulling on the thread really hard so that the knot becomes very small and tight. Because mathematical thread has no thickness, we can reduce a mathematical knot until it becomes a point and disappears. This continuous transformation is known as bachelors' unknotting. Introduction Set up Knot diagrams History Homotopy is no use for deforming knots because it allows a curve to pass through itself: all knots are homotopic to the trivial knot. We need to ensure that each ht is one-to-one. If ht has this property for all t 2 [0; 1] then h is called an isotopy. Ioannis Diamantis An introduction to Knot Theory To see why this is so, imagine pulling on the thread really hard so that the knot becomes very small and tight. Because mathematical thread has no thickness, we can reduce a mathematical knot until it becomes a point and disappears. This continuous transformation is known as bachelors' unknotting. Introduction Set up Knot diagrams History Homotopy is no use for deforming knots because it allows a curve to pass through itself: all knots are homotopic to the trivial knot. We need to ensure that each ht is one-to-one. If ht has this property for all t 2 [0; 1] then h is called an isotopy. Unfortunately, isotopy is no use for deforming knots either. Even though a knot cannot pass through itself under isotopy, all knots are isotopic to the trivial knot. Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History Homotopy is no use for deforming knots because it allows a curve to pass through itself: all knots are homotopic to the trivial knot. We need to ensure that each ht is one-to-one. If ht has this property for all t 2 [0; 1] then h is called an isotopy. Unfortunately, isotopy is no use for deforming knots either. Even though a knot cannot pass through itself under isotopy, all knots are isotopic to the trivial knot. To see why this is so, imagine pulling on the thread really hard so that the knot becomes very small and tight. Because mathematical thread has no thickness, we can reduce a mathematical knot until it becomes a point and disappears. This continuous transformation is known as bachelors' unknotting. Ioannis Diamantis An introduction to Knot Theory Introduction Set up Knot diagrams History Figure 9: Isotopy and Knots Ioannis Diamantis An introduction to Knot Theory Definition We say that a knot K1 is ambient isotopic to K2 if there exist a 3 3 3 3 continuous F : R × [0; 1] ! R such that F (_;t): R ! R is a homeomorphism, F (K; 0) = K1 and F (K; 1) = K2. Introduction Set up Knot diagrams History We need only make one minor modification to fix this problem: we just need to ensure that the space containing the knot moves continuously along with the knot and does not become irreparably contorted.
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