An Invitation to Knot Theory

Virtual and Classical

Heather A. Dye

• Publisher: Chapman and Hall/CRC
• Year: 2018
• Print ISBN: 9781498701648
• eBook ISBN: 9781315360096
• Edition: 1
• Topic: Mathematics
• Subtopic: Games in Mathematics
• Index: Mathematics

The Only Undergraduate Textbook to Teach Both Classical and Virtual Knot Theory

An Invitation to Knot Theory: Virtual and Classical gives advanced undergraduate students a gentle introduction to the field of virtual knot theory and mathematical research. It provides the foundation for students to research knot theory and read journal articles on their own. Each chapter includes numerous examples, problems, projects, and suggested readings from research papers. The proofs are written as simply as possible using combinatorial approaches, equivalence classes, and linear algebra.

The text begins with an introduction to virtual knots and counted invariants. It then covers the normalized f -polynomial (Jones polynomial) and other skein invariants before discussing algebraic invariants, such as the quandle and biquandle. The book concludes with two applications of virtual knots: textiles and quantum computation.

III

Algebraic structures

CHAPTER 11

Quandles

In this chapter, we introduce an algebraic structure named the quandle. A quandle consists of a set X and an operation ᐅ that satisfies algebraic versions of the Reidemeister moves. A quandle can be associated with an oriented virtual knot diagram. Equivalent oriented virtual knot diagrams are associated with the same quandle structure; the quandle is an invariant of oriented virtual knots. Invariants based on the quandle are defined using information from the classical crossings; a quandle invariant can distinguish between Kishino’s knot and its flip. This result negatively answers the question that we asked in Chapter 5: for all virtual knot diagrams K, is K ∼ K^F ?

11.1 Tricoloring

Before introducing quandles, we study tricoloring of virtual knot diagrams. Tricoloring a virtual knot diagram allows us to distinguish the trefoil (and other knots) from the unknot. (Tricoloring is actually an example of labeling a knot diagram with the elements of a quandle.) To tricolor a virtual knot diagram, we label (or color) the arcs of a knot diagram with elements of the set {r, g, b}. Recall that an arc of the diagram begins and terminates at the underpasses of classical crossings. In most of the literature on quandles, an arc is referred to as an edge. In this chapter, an edge begins and terminates at the underpasses.

To color a virtual knot diagram, we select an element from the set {r, g, b} for each edge in the diagram. Since we select edge labels or colors from a three element set, this is referred to as a tricoloring. The edge colors are selected using the following rules.

1. Each edge is labeled with an element of the set: {r, g, b}.

2. Each classical crossing is labeled so that either all three colors or exactly one color appears at the crossing.

In Figure 11.1, three virtual knot diagrams have been colored following these rules. The trefoil knot in Figure 11.1a is labeled with three colors, but it can also be lab...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• SECTION I Knots and crossings
• SECTION II Knot polynomials
• SECTION III Algebraic structures
• APPENDIX A: Tables
• APPENDIX B: References by chapter
• Index