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Surfaces Associated to a Space Curve: a New Proof of Fabricius-Bjerre's Formula

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Surfaces Associated to a Space Curve: a New Proof of Fabricius-Bjerre's Formula Surfaces associated to a space curve: A new proof of Fabricius-Bjerre's Formula vorgelegt von M. Sc. Barbara Jab lo´nska Kamie´nPomorski (Polen) Von der Fakult¨atII - Mathematik und Naturwissenschaften der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr.rer.nat. genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Fredi Tr¨oltzsch Berichter/Gutachter: Prof. John M. Sullivan, Ph.D. Berichter/Gutachter: Prof. Thomas Banchoff, Ph.D. Tag der wissenschaftlichen Aussprache: 23.05.2012 Berlin 2012 D 83 ii Abstract In 1962 Fabricius-Bjerre [16] found a formula relating certain geometric features of generic closed plane curves. Among bitangent lines, i.e., lines that are tangent to a curve at two points, distinguish two types: external - if the arcs of tangency lie on the same side of the line - and internal otherwise. Then, there is the following equality: the number of external bitangent lines equals the sum of the number of internal bitangent lines, the number of crossings and half of the number of inflection points of the plane curve. Two different proofs of the formula followed, by Halpern [20] in 1970 and Ban- choff [6] in 1974, as well as many generalizations. Halpern's approach is used here to provide a generalization from bitangent lines to parallel tangents pairs. The main result of this thesis is a new proof of Fabricius-Bjerre's Theorem, which uses new methods. The idea is to view a plane curve as a projection of a space curve. The proof establishes a connection between the generic plane curves and the projections of a certain class of generic closed space curves. For a closed space curve three surfaces are constructed with maps to the sphere of projection directions. These maps encode the information of the variables appearing in the Fabricius-Bjerre formula for each projection. The surfaces are glued together to form a closed surface with a naturally defined continuous map to the sphere. The degree of that map turns out to be equal to the expression arising when all variables of the Fabricius-Bjerre formula (for a plane curve which is a projection of the space curve) are put on one side of the equality. Using methods similar to those of V.I. Arnold (used to define J ± and St invariants), it is proved that the degree of the constructed map is zero for any generic space curve. An equivalence between the latter and Fabricius-Bjerre's Theorem is established. This gives a new proof of the theorem. The methods developed in this thesis have numerous possible applications. First, the integral version of the Fabricius-Bjerre formula is obtained which states that the average bitangency number equals the sum of the average crossing number and the average absolute torsion of a generic space curve. Another application is a new proof iii of the formula of Banchoff [5] and Aicardi [2] that expresses the self-linking number of a closed space curve as a sum of the writhe of a diagram and half of the signs of torsion at points of the space curve that project to inflection points in the diagram. It seems that using the methods of this work, nearly any formula relating some geometric features of a plane curve, as for example, the generalization to parallel tangents pairs, can be proved. Also, the generalization of the formula by Fabricius-Bjerre [17] to curves with cusps finds a new geometric interpretation with the new approach. iv To Krzy´sand Marysia vi Acknowledgments First of all this dissertation could not have been completed without the help of John M. Sullivan who not only served as an advisor but also encouraged and challenged the author throughout the entire academic program. The author wishes to thank John M. Sullivan for his continuous support and endless patience in helping her with tackling the details of the long proof. The author wants to express her appreciation for the many helpful suggestions and ideas which resulted from conversations with Francesca Aicardi, Thomas Banchoff, Elizabeth Denne, Emmanuel Ferrand, Gregor Masbaum, Clint McCrory, Renzo Ricca and many others who are not named here. Special thanks go to Jozef Przytycki for introducing the author to many great knot theorists and for inviting her to partic- ipate in the conference \Knots in Washington". The author wishes to thank Jason Cantarella for an invitation to complete a research stay at the University of Georgia (Athens) in the spring of 2010 and offering guidance which resulted in notable devel- opments in the author's research. Great acknowledgments go to mathematics school teachers Krystyna Tyc-Zochowska_ and Jacek Golusda who constantly challenged the author and thought her to love mathematics. Most of the figures in this thesis were generated with jReality. The author wishes to express her gratitude to all who supported her in the programming process, es- pecially to Martin Sommer who put the source code of his applet \Visualization in Geometric Knot Theory" at author's disposal, which eventually was the trigger to the entire research of this thesis. Also the visualization of the curves and surfaces would not be possible without technical support of Charles Gunn, Andr´eHeydt and Steffen Weißmann. For the constant support on the personal side the author wishes to thank her family and friends and especially a great mathematician Cristhian Emmanuel Garay. vii viii Contents List of Figures xi Introduction 1 I Closed plane curves 5 1 The Fabricius-Bjerre formula 6 1.1 Formulation by Halpern . .6 1.2 Proofs . .7 1.2.1 Fabricius-Bjerre's proof . .7 1.2.2 Halpern's proof . .9 1.2.3 Banchoff's proof . 12 1.3 Further versions and generalizations . 14 2 Formula for parallel tangents 19 3 Arnold's invariants versus Fabricius-Bjerre formula 28 3.1 Arnold's invariants . 28 3.2 A proof of Fabricius-Bjerre formula according to Banchoff's idea . 33 3.3 Formula for J ±-invariants. 35 II Closed space curves 39 4 Preliminaries 40 4.1 Notation . 40 4.2 General position . 42 ix 5 Reidemeister Curves 46 5.1 I curve . 49 5.2 II curve . 50 5.3 III curve . 70 6 Surfaces 74 6.1 Crossing surface . 82 6.2 Inflection surface . 90 6.3 Bitangency surface . 96 7 Surfaces related to Fabricius-Bjerre formula 107 8 A Self-Linking Number Formula 126 III A new proof of Fabricius-Bjerre's Theorem 137 9 The space C1(S1; R3) 138 9.1 Transversality Theorems . 138 9.1.1 Jet Bundles . 138 9.1.2 The Whitney C1-topology . 141 9.1.3 Transversality . 142 9.2 Fr´echet spaces . 143 9.2.1 Fr´echet spaces and manifolds . 144 9.2.2 Tame Fr´echet spaces. 148 10 The spaces of generic and non-generic curves 151 10.1 The set of generic space curves is open and dense . 151 10.2 Hypersurface of non-generic curves . 155 11 New proof of Fabricius-Bjerre's Theorem 162 11.1 Generic plane curves lift to generic space curves . 162 11.2 The degree map vanishes on G ...................... 169 12 Final remarks 175 Index of Symbols 180 Bibliography 181 x List of Figures 1 The Fabricius Bjerre formula. .1 2 A curve with 8 external and 1 internal bitangent pairs, 5 crossings and 4 inflection points. .2 1.1 The positive tangent ray passing through an external bitangent line of the second type (top) and through an inflection point (bottom). .8 1.2 The bitangency formula (1.4). .9 1.3 Possible moves that change the variables in the Fabricius-Bjerre formula. 13 1.4 An example of a curve with cusps. 14 1.5 Cusp of type 1 (left) and of type 2 (right). 15 1.6 The types of crossings and tangent-normal pairs of Thompson's formula. 18 2.1 External and internal parallel tangents pairs. 20 2.2 A negative and a positive parallel tangents pair involving an inflection point. ................................... 20 2.3 Attributes of a parallel tangents pair determining its sign. 21 2.4 An ellipse has two pairs of parallel tangents at angle π=2. The positive one is indicated in red (the horizontal pair of arrows) and the negative in blue (the vertical pair of arrows). 22 2.5 The pair (s; t) is a zero of the vector field W θ,φ............. 27 3.1 A direct and inverse self-tangency. 29 3.2 Perestroikas of an inverse (top) and a direct (bottom) self-tangency point. ................................... 29 3.3 A Perestroika of a triple point. 30 3.4 Two types of perestroika of a triple point according to the orientation of the vanishing triangle. 31 3.5 Standard curves of the Whitney index 0; ±1; ±2 ± 3; ±4; ::: ...... 31 3.6 An example of computation of Arnold's invariants from the definition. 33 xi 3.7 Examples of direct (red) and inverse (blue) bitangent pairs. 36 3.8 Values of the variables in (3.3) obtained from their definitions. 37 3.9 Jumps of the variables involved in (3.3) when passing through the strata of FB-discriminant. 38 4.1 Situations not allowed for a curve in general position by Definition 4.1 (ii): top (a) and bottom (b). Both of the situations are depicted from a random direction (left) and from direction γ(s) − γ(t) (right) at a pair fs; tg violating (ii). 43 4.2 A global singularity: curve is tangent to its tangent developable sur- face. (Here only a part of the tangent developable surface with λ ≥ 0 is pictured.) . 45 5.1 Three different orthogonal projections of the same space curve onto planes showing certain singularities.
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