Western Kentucky University TopSCHOLAR® Masters Theses & Specialist Projects Graduate School 5-2010 An Algorithm to Generate Two-Dimensional Drawings of Conway Algebraic Knots Jen-Fu Tung Western Kentucky University, [email protected] Follow this and additional works at: http://digitalcommons.wku.edu/theses Part of the Discrete Mathematics and Combinatorics Commons, and the Numerical Analysis and Computation Commons Recommended Citation Tung, Jen-Fu, "An Algorithm to Generate Two-Dimensional Drawings of Conway Algebraic Knots" (2010). Masters Theses & Specialist Projects. Paper 163. http://digitalcommons.wku.edu/theses/163 This Thesis is brought to you for free and open access by TopSCHOLAR®. It has been accepted for inclusion in Masters Theses & Specialist Projects by an authorized administrator of TopSCHOLAR®. For more information, please contact [email protected]. AN ALGORITHM TO GENERATE TWO-DIMENSIONAL DRAWINGS OF CONWAY ALGEBRAIC KNOTS A Thesis Presented to The Faculty of the Department of Mathematics and Computer Science Western Kentucky University Bowling Green, Kentucky In Partial Fulfillment Of the Requirements for the Degree Master of Science By Jen-Fu Tung May 2010 AN ALGORITHM TO GENERATE TWO-DIMENSIONAL DRAWINGS OF CONWAY ALGEBRAIC KNOTS Date Recommended __April 28, 2010___ __ Uta Ziegler ____________________ Director of Thesis __ Claus Ernst ____________________ __ Mustafa Atici __________________ ______________________________________ Dean, Graduate Studies and Research Date ACKNOWLEDGMENTS I would first like to acknowledge my thesis advisor, Dr. Uta Ziegler, for providing her guidance and great support on my studies of theory and programming in her lab since 2008. My writing, programming, and confidence have greatly improved under her supervision and encouragement. It has really been an honor to work with Dr. Ziegler. I would also like to express gratitude to my committee members, Dr. Claus Ernst and Dr. Mustafa Atici, for all the suggestions and corrections that made this thesis a solid, influential piece of work. I would like to show my appreciation for the faculty and staff of the Department of Mathematics and Computer Science at WKU for their help throughout my graduate research and study here. Special thanks to my parents and sister in Taiwan for their concerns and support to give me the strength to make this work possible. i TABLE OF CONTENTS CHAPTER 1: INTRODUCTION AND BASIC DEFINITIONS ....................................... 3 1.1 Introduction ............................................................................................................... 3 1.2 Basic definitions ........................................................................................................ 4 CHAPTER 2: RESEARCH ON GENERATING KNOT GRAPHS AND KNOT DRAWINGS ................................................................................................................ 11 2.1 Generating a planar map for a knot ......................................................................... 11 2.1.1 Generating a planar map for a knot using a rooted binary tree ........................ 11 2.1.2 Generating a planar map for a knot using a Hamiltonian cycle ....................... 18 2.2 Existing knot drawing algorithms ........................................................................... 20 CHAPTER 3: GENERATING CONWAY ALGEBRAIC KNOT DRAWINGS ............ 24 3.1 Conway algebraic knots and rooted binary trees .................................................... 24 3.2 Generating constructed planar maps using Conway algebraic trees ....................... 28 3.2.1 Generating a Conway algebraic tree ................................................................ 29 3.2.2 Generating a constructed planar map from a Conway algebraic tree .............. 35 3.3 Tree reduction operations ....................................................................................... 50 3.4 Flype operations ...................................................................................................... 73 CHAPTER 4: CONCLUSIONS ....................................................................................... 90 REFERENCES ................................................................................................................. 92 ii LIST OF FIGURES Figure 1.2.1. Crossings drawn in a regular planar projection. ............................................ 7 Figure 1.2.2. Four half-twists. ............................................................................................. 7 Figure 1.2.3. An example of a flype operation. Circles A and B are tangles. .................... 7 Figure 1.2.4. Two knots are alternating knots. (A) is a knot with three-crossings (also as known as the trefoil knot). (B) is a knot with seven-crossings....................................... 8 Figure 1.2.5. Forming a composite knot. ............................................................................ 8 Figure 1.2.6. Two types of Conway regions. ...................................................................... 9 Figure 1.2.7. An algebraic decomposition of a Conway algebraic knot. ............................ 9 Figure 2.1.1. A bridge graph that represents 1100011001. ............................................... 12 Figure 2.1.2. A bridge graph after adding an extra down edge. It represents 11000110010..................................................................................................................... 13 Figure 2.1.3. A new bridge graph with the positive prefix property obtained from Figure 2.1.2. The number in parentheses is the original number in Figure 2.1.2. ............ 15 Figure 2.1.4. A sequence pictures for a rooted binary tree generated by the binary string “111010000.” .................................................................................................................... 16 Figure 2.1.5. A rooted binary tree with additional edges added. ...................................... 17 Figure 2.1.6. Connecting new edges from internal vertices to the nearest leaf vertices. .. 17 Figure 2.1.7. The final connection for generating a rooted 4-regular planar map. ........... 18 Figure 2.1.8. The map drawn using a permutation vector {1, 3, 2, 5, 4, 2, 3, 6, 6, 4, 5, 7, 7, 1}. .................................................................................................................................. 19 Figure 2.1.9. An example to show the replacement of an intersecting edge. ................... 19 Figure 2.2.1. An example of circle packing. ..................................................................... 21 Figure 2.2.2. A 19-crossings planar graph generated by Mathematica. ........................... 21 Figure 2.2.3. Part of the Knotplot’s catalogue table. ........................................................ 22 Figure 2.2.4. A knot with three-crossings (the trefoil knot) generated by Knotplot. ........ 22 iii Figure 3.1.1. A Conway algebraic knot diagram decomposed by a set of Conway circles. ............................................................................................................................... 25 Figure 3.1.2. A Conway algebraic knot diagram with a set of Conway circles and its preferred shading. ............................................................................................................. 26 Figure 3.1.3. A CAD with a set of Conway circles and its preferred shading together with the corresponding preferred tree. The circles op1, op2, and op3 are referenced in section 3.2.1. ..................................................................................................................... 28 Figure 3.2.1. An example of the same rooted binary tree can be derived from two different Conway algebraic knot diagrams. ...................................................................... 30 Figure 3.2.2. The relationship of knots, Conway algebraic diagrams, and rooted binary trees. .................................................................................................................................. 30 Figure 3.2.3. RL overpass edge, LR overpass edge, and normal edge. ............................ 31 Figure 3.2.4. Possible arrangement of two Conway circles. Only (B) and (C) can cause an overpass edge. (A) shows a normal edge. .................................................................... 33 Figure 3.2.5. The relationship of knots, Conway algebraic diagrams, and CATs. ........... 33 Figure 3.2.6. A rooted binary tree generated from Figure 2.1.3. The name on each node is for use in Figure 3.2.7. .................................................................................................. 34 Figure 3.2.7. A screen shot of a CAT embedded by using the positive prefix binary string obtained from Figure 3.2.6 and added overpass type on some edges. A1 is the root of the displayed CAT. ....................................................................................................... 35 Figure 3.2.8. (A) is a simple box (dashed line); (B) is a combo box (dashed line) with two boxes (simple or combo) inside. ................................................................................ 37 Figure 3.2.9. We shade the upper side of the pair of type III Conway sub-regions in the combo box K′ for the internal node K. .............................................................................. 37 Figure 3.2.10. Pictures showing different rotations for a child box. (A) shows no rotation of box Pi′, (B) shows that an LR overpass edge