A brief Incursion into Knot Theory. Trinity University
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1 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
2 Outline 1 A Fundamental Problem (Balreira - Trinity University) Knot Theory Major Seminar 2 / 31
3 Outline 1 A Fundamental Problem 2 Knot Theory (Balreira - Trinity University) Knot Theory Major Seminar 2 / 31
4 Outline 1 A Fundamental Problem 2 Knot Theory 3 Reidemeister Moves (Balreira - Trinity University) Knot Theory Major Seminar 2 / 31
5 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
6 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
7 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
8 A Fundamental Problem Main Research Area Injectivity via Geometric and Topological Methods Foliation Theory (Balreira - Trinity University) Knot Theory Major Seminar 4 / 31
9 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
10 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
11 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
12 Sample Topics (Balreira - Trinity University) Knot Theory Major Seminar 5 / 31
13 Sample Topics Project Problem (Foliation Theory) Discuss Fibrations versus Foliations, e.g., characterize the foliations of the Euclidean Plane, that is, R 2. (Balreira - Trinity University) Knot Theory Major Seminar 5 / 31
14 Sample Topics Project Problem (Foliation Theory) Discuss Fibrations versus Foliations, e.g., characterize the foliations of the Euclidean Plane, that is, R 2. Project Problem (Spectral Theory) Discuss the eigenvalues of the Laplacian on a bounded region of R n, e.g., Rayleigh and Min-Max Methods. (Balreira - Trinity University) Knot Theory Major Seminar 5 / 31
15 Sample Topics (Balreira - Trinity University) Knot Theory Major Seminar 6 / 31
16 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
17 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) M (Balreira - Trinity University) Knot Theory Major Seminar 6 / 31
18 Knot Theory When can tell the difference between Knots? Are the knots below the same? (Balreira - Trinity University) Knot Theory Major Seminar 7 / 31
19 Knot Theory When can tell the difference between Knots? How about these? (Balreira - Trinity University) Knot Theory Major Seminar 8 / 31
20 Knot Theory Some Formalism A knot is an injective map h : S 1 R 3 Picture in the plane (or slide) - Diagram with crossing (Balreira - Trinity University) Knot Theory Major Seminar 9 / 31
21 Knot Theory Some Formalism A knot is an injective map h : S 1 R 3 Picture in the plane (or slide) - Diagram with crossing Tame Knots (Balreira - Trinity University) Knot Theory Major Seminar 9 / 31
22 Knot Theory Some Formalism A knot is an injective map h : S 1 R 3 Picture in the plane (or slide) - Diagram with crossing Tame Knots Finite Number of arcs (Balreira - Trinity University) Knot Theory Major Seminar 9 / 31
23 Knot Theory Some Formalism A knot is an injective map h : S 1 R 3 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
24 Knot Theory Some Formalism A knot is an injective map h : S 1 R 3 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
25 Knot Theory Some Formalism A knot is an injective map h : S 1 R 3 Picture in the plane (or slide) - Diagram with crossing Tame Knots Finite Number of arcs Only two strands at a crossing nice Invariant Property: K 1 K 2 if, (Balreira - Trinity University) Knot Theory Major Seminar 9 / 31
26 Knot Theory Some Formalism A knot is an injective map h : S 1 R 3 Picture in the plane (or slide) - Diagram with crossing Tame Knots Finite Number of arcs Only two strands at a crossing nice Invariant Property: K 1 K 2 if, There is a homeomorphism ϕ : R 3 R 3 such that ϕ(k 1 ) = K 2 (Balreira - Trinity University) Knot Theory Major Seminar 9 / 31
27 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
28 Knot Theory - Diagrams Two Knots with similar Diagram must be the same. Trefoil Braided Trefoil (Balreira - Trinity University) Knot Theory Major Seminar 11 / 31
29 Knot Theory - Diagrams Two Knots with similar Diagram must be the same. Trefoil Braided Trefoil (Balreira - Trinity University) Knot Theory Major Seminar 11 / 31
30 Knot Theory - Diagrams Two Knots with similar Diagram must be the same. Trefoil Braided Trefoil (Balreira - Trinity University) Knot Theory Major Seminar 11 / 31
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
32 Knot Theory - Diagrams Two Knots with similar Diagram must be the same. Trefoil Braided Trefoil (Balreira - Trinity University) Knot Theory Major Seminar 11 / 31
33 Knot Theory - Diagrams Even if not exactly the diagram same, they could be the same... (Balreira - Trinity University) Knot Theory Major Seminar 12 / 31
34 Moves that preserve the Knot Reidemeister Moves Type I: Put or Take out a kink. (Balreira - Trinity University) Knot Theory Major Seminar 13 / 31
35 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
36 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
37 Type I Put or Take out a kink. (Balreira - Trinity University) Knot Theory Major Seminar 14 / 31
38 Type II Slide a strand over/under to creat/remove two crossings. (Balreira - Trinity University) Knot Theory Major Seminar 15 / 31
39 Type III Slide a strand across a crossing (Balreira - Trinity University) Knot Theory Major Seminar 16 / 31
40 Unknotting a Knot Using Reidemeister moves (Balreira - Trinity University) Knot Theory Major Seminar 17 / 31
41 Unknotting a Knot Using Reidemeister moves (Balreira - Trinity University) Knot Theory Major Seminar 17 / 31
42 Unknotting a Knot Using Reidemeister moves (Balreira - Trinity University) Knot Theory Major Seminar 17 / 31
43 Unknotting a Knot Using Reidemeister moves (Balreira - Trinity University) Knot Theory Major Seminar 17 / 31
44 Knot Theory A Possible classification? (Balreira - Trinity University) Knot Theory Major Seminar 18 / 31
45 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
46 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
47 Invariants Colorability (Balreira - Trinity University) Knot Theory Major Seminar 19 / 31
48 Invariants Colorability Unknotting number (Balreira - Trinity University) Knot Theory Major Seminar 19 / 31
49 Invariants Colorability Unknotting number Genus of a knot (Balreira - Trinity University) Knot Theory Major Seminar 19 / 31
50 Invariants Colorability Unknotting number Genus of a knot Knot Group (Balreira - Trinity University) Knot Theory Major Seminar 19 / 31
51 Invariants Colorability Unknotting number Genus of a knot Knot Group Polynomial Invariants (Balreira - Trinity University) Knot Theory Major Seminar 19 / 31
52 Sample Problems in Knot Theory (Balreira - Trinity University) Knot Theory Major Seminar 20 / 31
53 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
54 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. Project Problem (Unknotting number) Discuss classical result and classification of knots via crossing and unknotting numbers, e.g., minimal diagrams and prime knots. (Balreira - Trinity University) Knot Theory Major Seminar 20 / 31
55 More Sample Problems in Knot Theory (Balreira - Trinity University) Knot Theory Major Seminar 21 / 31
56 More Sample Problems in Knot Theory Project Problem (Genus of a knot) Learn surface theory in order to understand Seifert surfaces. Show that the genus of a knot, χ(m) + 1 is a knot invariant. 2 (Balreira - Trinity University) Knot Theory Major Seminar 21 / 31
57 More Sample Problems in Knot Theory Project Problem (Genus of a knot) Learn surface theory in order to understand Seifert surfaces. Show that the genus of a knot, χ(m) + 1 is a knot invariant. 2 (Balreira - Trinity University) Knot Theory Major Seminar 21 / 31
58 Even More Sample Problems in Knot Theory Project Problem (Knot Group) Learn to compute the knot group and discuss group presentations and possibly learn about representation theory. (Balreira - Trinity University) Knot Theory Major Seminar 22 / 31
59 Even More Sample Problems in Knot Theory Project Problem (Knot Group) Learn to compute the knot group and discuss group presentations and possibly learn about representation theory. π 1 (R 3 T) = a,b aba = bab = x,y x 2 = y 3 Can we make sense of the expressions above? (Balreira - Trinity University) Knot Theory Major Seminar 22 / 31
60 One More Sample Problem in Knot Theory Project Problem (Polynomial Invariants) Understand and learn to compute a polynomial invariant of a knot, e.g., Bracket, Kauffman, Jones, Homfly, and Alexander Polynomials. All rules come from the topology of the knots (i.e., crossing types) (Balreira - Trinity University) Knot Theory Major Seminar 23 / 31
61 One More Sample Problem in Knot Theory Project Problem (Polynomial Invariants) Understand and learn to compute a polynomial invariant of a knot, e.g., Bracket, Kauffman, Jones, Homfly, and Alexander Polynomials. All rules come from the topology of the knots (i.e., crossing types) Some are formal Algebraic Computations. (Balreira - Trinity University) Knot Theory Major Seminar 23 / 31
62 One More Sample Problem in Knot Theory Project Problem (Polynomial Invariants) Understand and learn to compute a polynomial invariant of a knot, e.g., Bracket, Kauffman, Jones, Homfly, and Alexander Polynomials. All rules come from the topology of the knots (i.e., crossing types) Some are formal Algebraic Computations. Others are Geometric in nature. (Balreira - Trinity University) Knot Theory Major Seminar 23 / 31
63 Invariants Colorability K is colorable each arc has one of 3 colors At least two of the colors are used At crossing, either all different or all the same color. (Balreira - Trinity University) Knot Theory Major Seminar 24 / 31
64 Invariants Colorability K is colorable each arc has one of 3 colors At least two of the colors are used At crossing, either all different or all the same color. (Balreira - Trinity University) Knot Theory Major Seminar 24 / 31
65 Invariants Colorability K is colorable each arc has one of 3 colors At least two of the colors are used At crossing, either all different or all the same color. (Balreira - Trinity University) Knot Theory Major Seminar 24 / 31
66 Colorability The Unknot and Trefoil are different! Tricolorable Not tricolorable (Balreira - Trinity University) Knot Theory Major Seminar 25 / 31
67 An Algebraic Invariant Topologically, every knot is equivalent to S 1. DONE? (Balreira - Trinity University) Knot Theory Major Seminar 26 / 31
68 An Algebraic Invariant Topologically, every knot is equivalent to S 1. DONE? Knots are different via their embedding in R 3. (Balreira - Trinity University) Knot Theory Major Seminar 26 / 31
69 An Algebraic Invariant Topologically, every knot is equivalent to S 1. DONE? Knots are different via their embedding in R 3. Better question: K 1 K 2 if ϕ : R 3 R 3 with ϕ(k 1 ) = K 2 (Balreira - Trinity University) Knot Theory Major Seminar 26 / 31
70 An Algebraic Invariant Topologically, every knot is equivalent to S 1. DONE? Knots are different via their embedding in R 3. Better question: K 1 K 2 if ϕ : R 3 R 3 with ϕ(k 1 ) = K 2 Consider ϕ R 3 K 1 : R 3 K 1 R 3 K 2 (Balreira - Trinity University) Knot Theory Major Seminar 26 / 31
71 An Algebraic Invariant Topologically, every knot is equivalent to S 1. DONE? Knots are different via their embedding in R 3. Better question: K 1 K 2 if ϕ : R 3 R 3 with ϕ(k 1 ) = K 2 Consider ϕ R 3 K 1 : R 3 K 1 R 3 K 2 Thus π 1 (R 3 K 1 ) π 1 (R 3 K 2 ) (Balreira - Trinity University) Knot Theory Major Seminar 26 / 31
72 An Algebraic Invariant Topologically, every knot is equivalent to S 1. DONE? Knots are different via their embedding in R 3. Better question: K 1 K 2 if ϕ : R 3 R 3 with ϕ(k 1 ) = K 2 Consider ϕ R 3 K 1 : R 3 K 1 R 3 K 2 Thus π 1 (R 3 K 1 ) π 1 (R 3 K 2 ) Same Fundamental Group (Balreira - Trinity University) Knot Theory Major Seminar 26 / 31
73 The Knot Group An Algebraic Invariant π 1 (M) is the Fundamental Group of M (Balreira - Trinity University) Knot Theory Major Seminar 27 / 31
74 The Knot Group An Algebraic Invariant π 1 (M) is the Fundamental Group of M For a knot K, π 1 (R 3 K 1 ) is the Knot Group of K (Balreira - Trinity University) Knot Theory Major Seminar 27 / 31
75 The Knot Group An Algebraic Invariant π 1 (M) is the Fundamental Group of M For a knot K, π 1 (R 3 K 1 ) is the Knot Group of K MATH Topology (New Course! Fall 09) (Balreira - Trinity University) Knot Theory Major Seminar 27 / 31
76 The Knot Group An Algebraic Invariant π 1 (M) is the Fundamental Group of M For a knot K, π 1 (R 3 K 1 ) is the Knot Group of K MATH Topology (New Course! Fall 09) Some Algebra background would be very nice... (Balreira - Trinity University) Knot Theory Major Seminar 27 / 31
77 An Algebraic Invariant Types of Crossings + Crossing - Crossing (Balreira - Trinity University) Knot Theory Major Seminar 28 / 31
78 Computing the Knot group Label each arc with a letter. (Balreira - Trinity University) Knot Theory Major Seminar 29 / 31
79 Computing the Knot group Label each arc with a letter. At a crossing: the emerging arc labeled by the conjugate as follows: b b a c a c c = bab 1 c = b 1 ab (Balreira - Trinity University) Knot Theory Major Seminar 29 / 31
80 Computing the Knot group The Trefoil Group T Thus, b = aba 1 aab 1 a 1 or bab = aba (Balreira - Trinity University) Knot Theory Major Seminar 30 / 31
81 Computing the Knot group The Trefoil Group T a b c Thus, b = aba 1 aab 1 a 1 or bab = aba (Balreira - Trinity University) Knot Theory Major Seminar 30 / 31
82 Computing the Knot group The Trefoil Group T a b c Thus, b = aba 1 aab 1 a 1 or bab = aba (Balreira - Trinity University) Knot Theory Major Seminar 30 / 31
83 Computing the Knot group The Trefoil Group T a b b = cac 1 c Thus, b = aba 1 aab 1 a 1 or bab = aba (Balreira - Trinity University) Knot Theory Major Seminar 30 / 31
84 Computing the Knot group The Trefoil Group T a b a = bcb 1 c Thus, b = aba 1 aab 1 a 1 or bab = aba (Balreira - Trinity University) Knot Theory Major Seminar 30 / 31
85 Computing the Knot group The Trefoil Group T a b c c = aba 1 Thus, b = aba 1 aab 1 a 1 or bab = aba (Balreira - Trinity University) Knot Theory Major Seminar 30 / 31
86 Computing the Knot group The Trefoil Group T a b b = cac 1 a = bcb 1 c c = aba 1 Thus, b = aba 1 aab 1 a 1 or bab = aba (Balreira - Trinity University) Knot Theory Major Seminar 30 / 31
87 Computing the Knot group The Trefoil Group T a b b = cac 1 a = bcb 1 c c = aba 1 Thus, b = aba 1 aab 1 a 1 or bab = aba (Balreira - Trinity University) Knot Theory Major Seminar 30 / 31
88 Computing the Knot group The Trefoil Group The computations above show: (Balreira - Trinity University) Knot Theory Major Seminar 31 / 31
89 Computing the Knot group The Trefoil Group The computations above show: π 1 (R 3 T) = a,b aba = bab (Balreira - Trinity University) Knot Theory Major Seminar 31 / 31
90 Computing the Knot group The Trefoil Group The computations above show: π 1 (R 3 T) = a,b aba = bab Fact: π 1 (R 3 T) is not Abelian (Balreira - Trinity University) Knot Theory Major Seminar 31 / 31
91 Computing the Knot group The Trefoil Group The computations above show: π 1 (R 3 T) = a,b aba = bab Fact: π 1 (R 3 T) is not Abelian Fact: K is the unknot, π 1 (R 3 K) = Z (Balreira - Trinity University) Knot Theory Major Seminar 31 / 31
92 Computing the Knot group The Trefoil Group The computations above show: π 1 (R 3 T) = a,b aba = bab Fact: π 1 (R 3 T) is not Abelian Fact: K is the unknot, π 1 (R 3 K) = Z Unknot Trefoil! (Balreira - Trinity University) Knot Theory Major Seminar 31 / 31
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