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