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Knots, Braids and First Order Logic Siddhartha Gadgil and T. V. H. - PowerPoint PPT Presentation

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Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Knots, Braids and First Order Logic Siddhartha Gadgil and T. V. H. Prathamesh Indian Institute of Science,

  1. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Knots, Braids and First Order Logic Siddhartha Gadgil and T. V. H. Prathamesh Indian Institute of Science, Bangalore September 18, 2012 Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  2. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Outline Knots and Links 1 Link Axioms 2 Algebraic Formulation of Knot Theory 3 Stable Links and Infinite Braids 4 Infinite Braids as a Canonical Model 5 Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  3. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Knot Definition A knot K is defined as the image of a smooth, injective map h : S 1 → S 3 so that h ′ ( θ ) � = 0 for all θ ∈ S 1 . Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  4. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Knot Definition A knot K is defined as the image of a smooth, injective map h : S 1 → S 3 so that h ′ ( θ ) � = 0 for all θ ∈ S 1 . (Image source: Wikipedia) Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  5. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Link Definition A link L ⊂ S 3 is a smooth 1-dimensional submanifold of S 3 such that each component of L is a knot and there are only finitely many components. Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  6. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Link Definition A link L ⊂ S 3 is a smooth 1-dimensional submanifold of S 3 such that each component of L is a knot and there are only finitely many components. Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  7. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model When are two knots(or links) regarded as same or different? Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  8. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model When are two knots(or links) regarded as same or different? Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  9. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model When are two knots(or links) regarded as same or different? Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  10. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Definition (Ambient Isotopy) Two links L 1 and L 2 in S 3 are said to be ambient isotopic if there exists a smooth map F : S 3 × [0 , 1] → S 3 such that Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  11. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Definition (Ambient Isotopy) Two links L 1 and L 2 in S 3 are said to be ambient isotopic if there exists a smooth map F : S 3 × [0 , 1] → S 3 such that 1 F | S 3 ×{ 0 } = id | S 3 : S 3 → S 3 . Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  12. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Definition (Ambient Isotopy) Two links L 1 and L 2 in S 3 are said to be ambient isotopic if there exists a smooth map F : S 3 × [0 , 1] → S 3 such that 1 F | S 3 ×{ 0 } = id | S 3 : S 3 → S 3 . 2 F | S 3 ×{ 1 } ( L 1 ) = L 2 . Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  13. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Definition (Ambient Isotopy) Two links L 1 and L 2 in S 3 are said to be ambient isotopic if there exists a smooth map F : S 3 × [0 , 1] → S 3 such that 1 F | S 3 ×{ 0 } = id | S 3 : S 3 → S 3 . 2 F | S 3 ×{ 1 } ( L 1 ) = L 2 . 3 F | S 3 ×{ t } is a diffeomorphism ∀ t ∈ [0 , 1] Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  14. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Definition (Ambient Isotopy) Two links L 1 and L 2 in S 3 are said to be ambient isotopic if there exists a smooth map F : S 3 × [0 , 1] → S 3 such that 1 F | S 3 ×{ 0 } = id | S 3 : S 3 → S 3 . 2 F | S 3 ×{ 1 } ( L 1 ) = L 2 . 3 F | S 3 ×{ t } is a diffeomorphism ∀ t ∈ [0 , 1] 4 F is smooth. Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  15. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Definition (Ambient Isotopy) Two links L 1 and L 2 in S 3 are said to be ambient isotopic if there exists a smooth map F : S 3 × [0 , 1] → S 3 such that 1 F | S 3 ×{ 0 } = id | S 3 : S 3 → S 3 . 2 F | S 3 ×{ 1 } ( L 1 ) = L 2 . 3 F | S 3 ×{ t } is a diffeomorphism ∀ t ∈ [0 , 1] 4 F is smooth. Ambient isotopy induces an equivalence relation between links. Knot Equivalence Problem : Given two knots K 1 and K 2 , are they ambient isotopic to each other? Unknotting Problem : Given two knot, is it ambient isotopic to the unknot? Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  16. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Stable Equivalence of Links Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  17. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Stable Equivalence of Links Definition A link L ′ is said to be a stabilisation of a link L if the following conditions hold. Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  18. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Stable Equivalence of Links Definition A link L ′ is said to be a stabilisation of a link L if the following conditions hold. 1 L ′ = L ∪ L ′′ with L ′′ disjoint from L . Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  19. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Stable Equivalence of Links Definition A link L ′ is said to be a stabilisation of a link L if the following conditions hold. 1 L ′ = L ∪ L ′′ with L ′′ disjoint from L . 2 There is a collection of disjoint, smoothly embedded discs n D = { D 1 , D 2 , . . . , D n } in S 3 \ L , with L ′′ = � ∂ D i i =1 Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  20. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Stable Equivalence of Links Definition A link L ′ is said to be a stabilisation of a link L if the following conditions hold. 1 L ′ = L ∪ L ′′ with L ′′ disjoint from L . 2 There is a collection of disjoint, smoothly embedded discs n D = { D 1 , D 2 , . . . , D n } in S 3 \ L , with L ′′ = � ∂ D i i =1 Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  21. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Stable Equivalence of Links Definition A link L ′ is said to be a stabilisation of a link L if the following conditions hold. 1 L ′ = L ∪ L ′′ with L ′′ disjoint from L . 2 There is a collection of disjoint, smoothly embedded discs n D = { D 1 , D 2 , . . . , D n } in S 3 \ L , with L ′′ = � ∂ D i i =1 Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

  22. Knots and Links Link Axioms Algebraic Formulation of Knot Theory Stable Links and Infinite Braids Infinite Braids as a Canonical Model Stable Equivalence of Links Definition A link L ′ is said to be a stabilisation of a link L if the following conditions hold. 1 L ′ = L ∪ L ′′ with L ′′ disjoint from L . 2 There is a collection of disjoint, smoothly embedded discs n D = { D 1 , D 2 , . . . , D n } in S 3 \ L , with L ′′ = � ∂ D i i =1 Siddhartha Gadgil and T. V. H. Prathamesh Knots, Braids and First Order Logic

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