Knots, links, spatial graphs, and algebraic invariants : AMS special session on algebraic and combinatorial structures in knot theory, October 24-25, 2015, California State University, Fullerton, CA : AMS special session on spatial graphs, October 24-25,

This volume contains the proceedings of the AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory and the AMS Special Session on Spatial Graphs, both held from October 24–25, 2015, at California State University, Fullerton, CA. Included in this volume are articles that draw on techniques from geometry and algebra to address topological problems about knot theory and spatial graph theory, and their combinatorial generalizations to equivalence classes of diagrams that are preserved under a set of Reidemeister-type moves. The interconnections of these areas and their connections within the broader field of topology are illustrated by articles about knots and links in spatial graphs and symmetries of spatial graphs in $S^3$ and other 3-manifolds. Cover 1 Title page 2 Contents 4 Preface: Knots, graphs, algebra & combinatorics 6 Part I: Knot Theoretic Structures 6 Part II: Spatial Graph Theory 7 References 8 The first coefficient of Homflypt and Kauffman polynomials: Vertigan proof of polynomial complexity using dynamic programming 10 1. Introduction 10 2. Computation of P0(a) 11 3. Computation of P2i(a). 12 4. Coefficients of the Kauffman polynomial, F_{L}(a,z). 13 5. Polynomials of virtual diagrams. 14 6. Dynamic programming 14 7. Knotoids of Vladimir Turaev 15 8. Acknowledgements 15 References 15 Linear Alexander quandle colorings and the minimum number of colors 16 1. Introduction 16 2. Review of Quandles 17 3. Coloring of Knots by Linear Alexander Quandles of order 5 17 4. Main Result 19 5. Four Colors is the Minimum Number of Colors 28 References 30 Quandle identities and homology 32 1. Introduction 32 2. Preliminary 33 3. Type 3 quandles 34 4. From identities to extensions and subcomplexes 35 5. Inner identities 40 Acknowledgements 43 References 43 Ribbonlength of folded ribbon unknots in the plane 46 1. Introduction 46 2. Modeling Folded Ribbon Knots 47 3. Ribbon Equivalence 50 4. Ribbonlength 52 5. Local structure of folded ribbon knots 54 6. Projection stick index and ribbonlength 57 Acknowledgments 59 References 59 Checkerboard framings and states of virtual link diagrams 62 1. Introduction 62 2. Virtual knots 63 3. The Bracket polynomial 65 4. Establishing the result 71 5. Conclusion 72 References 72 Virtual covers of links II 74 1. Background 75 2. Semi-Fibered Concordance 77 3. Ribbon and Slice Obstructions 80 4. Injectivity of Satellite Operators 82 5. Concordance and Cables of Knots in 3-manifolds 85 Acknowledgments 87 References 88 Recent developments in spatial graph theory 90 1. Introduction 90 2. Intrinsic linking and knotting 91 3. n-apex graphs 93 4. Conway-Gordon type theorems for graphs in F(K6) and F(K7) 95 5. Conway-Gordon type theorems for K_{3,3,1,1} 97 6. Linear embeddings of graphs 98 7. Symmetries of spatial graphs in S3 102 8. Graphs embedded in 3-Manifolds 107 References 108 Order nine MMIK graphs 112 Introduction 112 1. Definitions and Lemmas 113 2. Proof of Proposition 2 115 3. Proof of Proposition 4 116 4. Computer Verification for Size 23 through 27 129 Acknowledgements 132 References 132 A chord graph constructed from a ribbon surface-link 134 1. Introduction 134 2. How to transform a (welded virtual) link diagram into a chord diagram without base crossing 137 3. How to transform a chord graph into a ribbon surface-link in 4-space 141 4. How to modify the moves on a chord diagram into the moves on a chord diagram without base crossing 142 References 145 The K_{n+5} and K_{32,1n} families and obstructions to n-apex. 146 Introduction 146 1. Proof of Theorem 3 148 2. Results for \On{2} and \On{3} 154 Appendix A. Edge lists of K8 family graphs 156 Appendix B. K8 family graphs are not 3-apex 156 Appendix C. Proper minors are 3-apex 156 References 167 Partially multiplicative biquandles and handlebody-knots 168 1. Introduction 168 2. Biquandles 169 3. n-Parallel Biquandles 171 4. G-Families of Biquandles 174 5. Partially Multiplicative Biquandles 176 6. Counting Invariants 181 7. Questions 184 References 184 Tangle insertion invariants for pseudoknots, singular knots, and rigid vertex spatial graphs 186 1. Introduction 186 2. Tangle Insertion Invariants 187 3. Examples 191 4. Questions & Future Work 196 Acknowledgements 197 References 197 Back Cover 202