13.1 Introduction to 3-manifolds and their relationship to knots

Three-dimensional manifolds, or 3-manifolds, are spaces that look like our everyday 3D world up close. They're the playground for knots, those loopy curves we study in knot theory. Understanding 3-manifolds helps us grasp how knots behave in different spaces.

Knots can be embedded in various 3-manifolds, not just the usual 3D space we're used to. This affects how knots interact with their surroundings. The relationship between knots and 3-manifolds reveals deep insights about both, shaping our understanding of topology.

Introduction to 3-Manifolds

Basics of 3-manifolds

• A 3-manifold is a topological space that locally resembles Euclidean 3-space ($\mathbb{R}^3$) meaning every point has a neighborhood homeomorphic to an open subset of $\mathbb{R}^3$
• 3-manifolds are three-dimensional spaces without boundaries that can be thought of as generalizations of surfaces to higher dimensions
• Examples of 3-manifolds include $\mathbb{R}^3$ (Euclidean 3-space), $S^3$ (3-sphere), and $S^1 \times S^2$ (product of a circle and a 2-sphere)
• 3-manifolds can be orientable or non-orientable
  • Orientable 3-manifolds have a consistent choice of orientation (handedness) throughout the space like a sphere or torus
  • Non-orientable 3-manifolds, such as the 3-dimensional real projective space ($\mathbb{RP}^3$), do not have a consistent orientation and contain Möbius band-like structures

Knot embedding in 3-manifolds

• A knot is a closed, non-self-intersecting curve in a 3-manifold defined mathematically as an embedding of the circle ($S^1$) into the 3-manifold
• Knots can be embedded in various 3-manifolds, such as $\mathbb{R}^3$ (the most common setting for studying knots), $S^3$ (obtained by compactifying $\mathbb{R}^3$ with a point at infinity), and other 3-manifolds like lens spaces or Seifert-fibered spaces
• The embedding of a knot in a 3-manifold is called a knot type or knot class, and knots are considered equivalent if there exists an ambient isotopy between them within the 3-manifold

Relationship between Knots and 3-Manifolds

Knots and 3-manifolds relationship

• The complement of a knot in a 3-manifold is the 3-manifold obtained by removing the knot and its interior, for example, the knot complement of a knot $K$ in $S^3$ is $S^3 \setminus K$
• The knot complement is a 3-manifold with boundary, where the boundary is a torus ($S^1 \times S^1$), and it encodes important information about the knot, such as its fundamental group (knot group), Alexander polynomial, and Jones polynomial
• Surgery on a knot in a 3-manifold produces a new 3-manifold through a process called Dehn surgery, which involves removing a tubular neighborhood of the knot and gluing in a solid torus with different surgery coefficients resulting in different 3-manifolds

Topology of knot-containing 3-manifolds

• The genus of a knot is the minimum genus of any orientable surface bounded by the knot in the 3-manifold and serves as a measure of the knot's complexity related to the topology of the 3-manifold
• The Thurston-Bennequin number and the rotation number are invariants of Legendrian knots in contact 3-manifolds that provide information about the contact structure and the knot's embedding
• The JSJ decomposition (Jaco-Shalen-Johannson decomposition) decomposes a 3-manifold into simpler pieces along incompressible tori, which can be used to study the structure of 3-manifolds containing knots
• The geometrization theorem, proved by Perelman, states that every closed, orientable 3-manifold can be decomposed into geometric pieces, and the geometry of these pieces is related to the properties of knots embedded in the 3-manifold