1. Introduction to space curves and knot theory

Chapter 1 – Introduction to space curves and knot theory

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Introduction to space curves and knot theory

1.1 Fundamentals of space curves: homeomorphism and elementary curve in Euclidean three-dimensional space, regularity and smoothness, parametric representation, intrinsic Frénet frame, curvature and torsion, fundamental theorem, Frénet-Serret equations. 1.2 Global geometric aspects of space curves: integral quantities, Sherrer theorem, winding number, rotation index, Fenchel theorem, knot in three-dimensions, FaryMilnor theorem 1.3 Knots, links and projections: standard projection, over- and under-crossing sign convention, topological invariant, knot type, n-component link type, minimum number of crossings, linking number, topological classification, Reidemeister moves. 1.4 Gauss linking number: Gauss linking number, Gauss map, spherical indicatrix, solid angle interpretation, linking number of n-component link type. 1.5 Calugareanu-White invariant: ribbon, writhing number, total twist, intrinsic twist, self-linking invariant, fundamental properties of self-linking, writhing and total twist number. 1.6 Measures of structural complexity: global geometric quantities (coiling, packing, etc.), topological measures (linking numbers, crossing numbers, etc.), algebraic measures (average crossing number, etc.), estimated measures, tangle complexity. 1.7 Articles included: Ricca, R.L. (2005) Knot Theory. Structural complexity. In Encyclopedia of Nonlinear Science (ed. A. Scott), pp. 499-501, 885-887. Routledge, New York and London. Hoste, J., Thistlethwaite M. & Weeks, J. (1998). The first 1,701,936 knots. Math. Intelligencer 20, 33-48. Further reading: a good introduction to differential geometry is provided by: do Carmo, M.P. 1976 Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs, New Jersey. An introduction to geometric topology is given by: Milnor, J. 1969 Topology from Differential Viewpoint. The University Press of Virginia. A good introduction to knot theory is given by: Adams, C. 1994. The Knot Book,. W.H. Freeman Publisher, New York.

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Chapter 1 – Introduction to space curves and knot theory

Beginning of topological fluid mechanics: Lord Kelvin drawings of knots and braids taken from different pages of his personal notebooks. Note the tentative coding of the braid pattern (bottom, right-hand-side).

Modern applied topology: Hopf link, figure-of-eight knot and Whitehead link catenanes produced by topoisomerase II enzyme (from: Stasiak & Koller, 1988. In Fractals, Quasicrystals, Chaos, Knots and Algebraic Quantum Mechanics, ed. A. Aman, Kluwer).

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Chapter 1 – Introduction to space curves and knot theory

Computerized topological fluid mechanics: pattern of streamlines in spherical shell colorcoded according to intensity. Note the null points marked by the red dots (from Kitauchi et al., RIMS Kyoto, Phys. Today 12, 1996).

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Chapter 1 – Introduction to space curves and knot theory

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Chapter 1 – Introduction to space curves and knot theory

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Chapter 1 – Introduction to space curves and knot theory

1.1 Fundamentals of space curves

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Chapter 1 – Introduction to space curves and knot theory

1.2 Global geometric aspects of space curves

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Chapter 1 – Introduction to space curves and knot theory

1.3 Knots, links and projections

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1.4 Gauss linking number

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1.5 Calugareanu-White invariant

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1.6 Measures of structural complexity

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