TIGHTENING CURVES AND GRAPHS ON SURFACES BY HSIEN-CHIH CHANG DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science in the Graduate College of the University of Illinois at Urbana-Champaign, 2018 Urbana, Illinois Doctoral Committee: Professor Jeff Erickson, Chair Professor Chandra Chekuri 1 Professor Nathan M. Dunfield Professor David Eppstein Associate Professor Alexandra Kolla ii 2 R 3 ∆ 4 {f ª愛的88½½ 5 謝謝`們的愛 iii iv Abstract 6 Any continuous deformation of closed curves on a surface can be decomposed into a finite sequence of local changes 7 on the structure of the curves; we refer to such local operations as homotopy moves. Tightening is the process of 8 deforming given curves into their minimum position; that is, those with minimum number of self-intersections. 9 While such operations and the tightening process has been studied extensively, surprisingly little is known about 10 the quantitative bounds on the number of homotopy moves required to tighten an arbitrary curve. 11 An unexpected connection exists between homotopy moves and a set of local operations on graphs called 12 electrical transformations. Electrical transformations have been used to simplify electrical networks since the 19th 13 century; later they have been used for solving various combinatorial problems on graphs, as well as applications in 14 statistical mechanics, robotics, and quantum mechanics. Steinitz, in his study of 3-dimensional polytopes, looked 15 at the electrical transformations through the lens of medial construction, and implicitly established the connection 16 to homotopy moves; later the same observation has been discovered independently in the context of knots. 17 In this thesis, we study the process of tightening curves on surfaces using homotopy moves and their con- 18 sequences on electrical transformations from a quantitative perspective. To derive upper and lower bounds we 19 utilize tools like curve invariants, surface theory, combinatorial topology, and hyperbolic geometry. We develop 20 several new tools to construct efficient algorithms on tightening curves and graphs, as well as to present examples 21 where no efficient algorithm exists. We then argue that in order to study electrical transformations, intuitively it is 22 most beneficial to work with monotonic homotopy moves instead, where no new crossings are created throughout 23 the process; ideas and proof techniques that work for monotonic homotopy moves should transfer to those for 24 electrical transformations. We present conjectures and partial evidence supporting the argument. v vi Acknowledgments 25 First I would like to express my most sincere gratitude towards my Ph.D. advisor, Jeff Erickson, for your guidance 26 throughout the years. It is an honor to be your student. You always give me the most honest advice; every meeting 27 with you is fun and fruitful. Most of all, your patience and listening ears lighten the burden when life was difficult 28 and research was slow. I am forever grateful to know you as a colleague, and a friend. 29 It is always joyful to talk to Sariel Har-Peled, either about research or just random fun. Thank you for 30 introducing me around on my first day in school, and all the ideas and advice (and jokes!) you shared with me. I 31 want to thank Alexandra Kolla for your constant support and trust. Thank you for always giving me opportunities 32 to travel and collaborate, even when no progress has been made for a long time. 33 The theory group in University of Illinois at Urbana-Champaign provides the most healthy research environment 34 for me to grow as a young researcher. The collaborative atmosphere allows us to explore freely among the topics 35 that interest us. I am fortunate to have so many great colleagues to work and share experience with, and sometimes 36 just having fun together in the last few wonderful years: Shashank Agrawal, Matthew Bauer, Shant Boodaghians, 37 Charles Carlson, Timothy Chan, Karthik Chandrasekaran, Chandra Chekuri, Kyle Fox, Shamoli Gupta, Mitchell 38 Jones, Konstantinos Koiliaris, Nirman Kumar, Patrick Lin, Vivek Madan, Sahand Mozaffari, Kent Quanrud, Benjamin 39 Raichel, Tasos Sidiropulos, Matus Telgarsky, Ross Vasko, Yipu Wang, Chao Xu, Ching-Hua Yu, Xilin Yu, and many 40 more that I didn’t have the chance to know better. 41 I am very fortunate to have the chance of working with great people all around the world: Paweł Gawrychowski, 42 Naonori Kakimura, David Letscher, Arnaud de Mesmay, Shay Mozes, Saul Schleimer, Eric Sedgwick, Dylan Thurston, 43 Stephan Tillmann, and Oren Weimann. Working together with similar minds on interesting projects is much more 44 fun than working alone. Also, I want to express my thanks to Erin Chambers, Ho-Lin Chen, Kai-Min Chung, Joshua 45 Grochow, Chung-Shou Liao, Michael Pelsmajer, and Marcus Schaefer for providing me opportunities to travel and 46 discuss research with you. 47 Without the days working with friends in the theory group of National Taiwan University, I might not even 48 pursue the career of being a theoretical computer scientist. Thanks to Hsueh-I Lu for your inspiring lectures, 49 guidance, and advice on research that lead me to where I am. 50 最後，y%感謝»PHH與昕R；`們/我生}-的喜樂。與³攜K同L使我知S，愛/最'的g秘。 51 Finally, special thanks to Chichi and Luke; you are the joy of my life. Being with you made me realize that love is 52 the greatest mystery. vii 53 Table of Contents 54 Chapter 1 Introduction and History1 55 1.1 Homotopy Moves................................................... 1 56 1.2 Electrical Transformations............................................. 3 57 1.3 Relation between Two Local Operations .................................... 4 58 1.4 Results and Outline of the Thesis ........................................ 5 59 1.5 History and Related Work............................................. 5 60 1.6 Acknowledgment .................................................. 7 61 Chapter 2 Preliminaries 9 62 2.1 Surfaces........................................................ 9 63 2.2 Curves and Graphs on Surfaces.......................................... 9 64 2.2.1 Curves .................................................... 9 65 2.2.2 Graphs and Their Embeddings ..................................... 10 66 2.2.3 Curves as 4-regular Maps......................................... 10 67 2.2.4 Jordan Curve Theorem........................................... 11 68 2.2.5 Multicurves .................................................. 11 69 2.2.6 Tangles..................................................... 11 70 2.3 Homotopy ...................................................... 12 71 2.3.1 Covering Spaces and Fundamental Groups.............................. 12 72 2.3.2 Lifting .................................................... 12 73 2.4 Combinatorial Properties of Curves....................................... 13 74 2.4.1 Monogons and Bigons........................................... 13 75 2.4.2 Homotopy Moves ............................................. 13 76 2.4.3 Signs and Winding Numbers....................................... 13 77 2.5 Relating Graphs to Curves............................................. 14 78 2.5.1 Medial Construction............................................ 14 79 2.5.2 Facial Electrical Moves .......................................... 14 80 2.5.3 Depths of Planar and Annular Multicurves.............................. 15 81 2.5.4 Smoothing.................................................. 15 82 2.6 Tightening Curves via Bigon Removal...................................... 16 83 Chapter 3 Curve Invariant — Defect 17 84 3.1 Defect Lower Bound ................................................ 18 85 3.1.1 Flat Torus Knots .............................................. 18 86 3.1.2 Defects of arbitrary flat torus knots................................... 20 87 3.2 Defect Upper Bound ................................................ 23 88 3.2.1 Winding Numbers and Diameter .................................... 23 89 3.2.2 Inclusion-Exclusion ............................................ 24 90 3.2.3 Divide and Conquer............................................ 27 91 3.3 Medial Defect is Independent of Planar Embeddings............................. 29 92 3.3.1 Navigating Between Planar Embeddings ............................... 29 93 3.3.2 Tangle Flips................................................. 30 94 3.4 Implications for Random Knots........................................... 31 viii 95 Chapter 4 Lower Bounds for Tightening Curves 33 96 4.1 Lower Bounds for Planar Curves......................................... 33 97 4.1.1 Multicurves................................................. 34 98 4.2 Quadratic Bound for Curves on Surfaces.................................... 36 99 4.3 Quadratic Bound for Contractible Curves on Surfaces............................ 38 100 4.3.1 Traces and Types.............................................. 38 101 4.3.2 A Bad Contractible Annular Curve ................................... 39 102 4.3.3 More Complicated Surfaces ....................................... 40 103 Chapter 5 Tightening Planar Curves 43 104 5.1 Planar Curves .................................................... 43 105 5.1.1 Contracting Simple Loops ........................................ 43 106 5.1.2 Tangles.................................................... 44 107 5.1.3 Main Algorithm............................................... 46 108 5.1.4 Efficient Implementation........................................