Introduction
Knot theory is a branch of mathematics that studies the properties and classifications of knots, which are closed loops in three-dimensional space. Knots can be imagined as pieces of string that are tied in various ways and then joined at the ends. Two knots are considered equivalent if they can be transformed into each other by moving them around without cutting or passing through themselves. Knot theory explores how to distinguish different knots, how to measure their complexity, and how they relate to other mathematical objects such as surfaces, groups, and polynomials.
Contents
Knots and Links
Knots and Links
Tangles and Categories of Tangles
Knot and Link Invariants
Grading and Filtration
Singular Knots and Vassiliev Invariants
Chord Diagrams
Universal Quantum Invariant
The Kontsevich Integral
Formality Maps for Tangles
Braids
Braids and Braids Group
The Alexander and Markov Theorems
Pure Braids
Garside Normal Form