**Contemporary Mathematics**

Volume: 689;
2017;
189 pp;
Softcover

MSC: Primary 05; 57;

Print ISBN: 978-1-4704-2847-1

Product Code: CONM/689

List Price: $111.00

AMS Member Price: $88.80

MAA member Price: $99.90

**Electronic ISBN: 978-1-4704-4077-0
Product Code: CONM/689.E**

List Price: $111.00

AMS Member Price: $88.80

MAA member Price: $99.90

# Knots, Links, Spatial Graphs, and Algebraic Invariants

Share this page *Edited by *
*Erica Flapan; Allison Henrich; Aaron Kaestner; Sam Nelson*

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.

#### Readership

Graduate students and research mathematicians interested in knot theory.

# Table of Contents

## Knots, Links, Spatial Graphs, and Algebraic Invariants

- Cover Cover11
- Title page i2
- Contents iii4
- Preface: Knots, graphs, algebra & combinatorics v6
- The first coefficient of Homflypt and Kauffman polynomials: Vertigan proof of polynomial complexity using dynamic programming 110
- 1. Introduction 110
- 2. Computation of ๐โ(๐) 211
- 3. Computation of ๐โแตข(๐). 312
- 4. Coefficients of the Kauffman polynomial, ๐น_{๐ฟ}(๐,๐ง). 413
- 5. Polynomials of virtual diagrams. 514
- 6. Dynamic programming 514
- 7. Knotoids of Vladimir Turaev 615
- 8. Acknowledgements 615
- References 615

- Linear Alexander quandle colorings and the minimum number of colors 716
- Quandle identities and homology 2332
- Ribbonlength of folded ribbon unknots in the plane 3746
- Checkerboard framings and states of virtual link diagrams 5362
- Virtual covers of links II 6574
- Recent developments in spatial graph theory 8190
- 1. Introduction 8190
- 2. Intrinsic linking and knotting 8291
- 3. ๐-apex graphs 8493
- 4. Conway-Gordon type theorems for graphs in โฑ(๐ฆโ) and โฑ(๐ฆโ) 8695
- 5. Conway-Gordon type theorems for ๐พ_{3,3,1,1} 8897
- 6. Linear embeddings of graphs 8998
- 7. Symmetries of spatial graphs in ๐ยณ 93102
- 8. Graphs embedded in 3-Manifolds 98107
- References 99108

- Order nine MMIK graphs 103112
- A chord graph constructed from a ribbon surface-link 125134
- 1. Introduction 125134
- 2. How to transform a (welded virtual) link diagram into a chord diagram without base crossing 128137
- 3. How to transform a chord graph into a ribbon surface-link in 4-space 132141
- 4. How to modify the moves on a chord diagram into the moves on a chord diagram without base crossing 133142
- References 136145

- The ๐พ_{๐+5} and ๐พ_{3ยฒ,1โฟ} families and obstructions to ๐-apex. 137146
- Partially multiplicative biquandles and handlebody-knots 159168
- Tangle insertion invariants for pseudoknots, singular knots, and rigid vertex spatial graphs 177186
- Back Cover Back Cover1202