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1. Introduction to space curves and knot theory

1.1 Fundamentals of space curves: homeomorphism and elementary curve in Euclideanthree-dimensional space, regularity and smoothness, parametric representation,intrinsic Frénet frame, curvature and torsion, fundamental theorem, Frénet-Serretequations.

1.2 Global geometric aspects of space curves: integral quantities, Sherrer theorem,winding number, rotation index, Fenchel theorem, knot in three-dimensions, Fary-Milnor theorem

1.3 Knots, links and projections: standard projection, over- and under-crossing signconvention, topological invariant, knot type, n-component link type, minimumnumber of crossings, linking number, topological classification, Reidemeister moves.

1.4 Gauss linking number: Gauss linking number, Gauss map, spherical indicatrix, solidangle 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 twistnumber.

1.6 Measures of structural complexity: global geometric quantities (coiling, packing,etc.), topological measures (linking numbers, crossing numbers, etc.), algebraicmeasures (average crossing number, etc.), estimated measures, tangle complexity.

1.7 Articles included:Ricca, R.L. (2005) Knot Theory. Structural complexity. In Encyclopedia of Non-

linear Science (ed. A. Scott), pp. 499-501, 885-887. Routledge, New York andLondon.

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

Modern applied topology: Hopf link, figure-of-eight knot and Whitehead link catenanesproduced 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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Computerized topological fluid mechanics: pattern of streamlines in spherical shell color-coded according to intensity. Note the null points marked by the red dots (from Kitauchiet al., RIMS Kyoto, Phys. Today 12, 1996).

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1.1 Fundamentals of space curves

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1.2 Global geometric aspects of space curves

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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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