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.

part 1|75 pages

Knots and crossings

chapter 1|16 pages

Virtual knots and links

chapter 2|16 pages

Linking invariants

chapter 3|14 pages

A multiverse of knots

chapter 4|14 pages

Crossing invariants

chapter 5|13 pages

Constructing knots

part 2|77 pages

Knot polynomials

chapter 6|15 pages

The bracket polynomial

chapter 7|19 pages


chapter 8|11 pages

Bracket polynomial II

chapter 9|12 pages

The checkerboard framing

chapter 10|15 pages

Modifications of the bracket polynomial

part 3|65 pages

Algebraic structures

chapter 11|12 pages


chapter 12|16 pages

Knots and quandles

chapter 13|13 pages


chapter 14|14 pages

Gauss diagrams

chapter 15|7 pages