🪢Knot Theory Unit 4 – The Fundamental Group and the Knot Group

Key Concepts and Definitions

• The fundamental group is a topological invariant that captures the essential structure of a topological space
• Homotopy is an equivalence relation between continuous functions that can be continuously deformed into each other
• Path homotopy is a homotopy between paths with fixed endpoints
  • Two paths are path homotopic if they can be continuously deformed into each other while keeping the endpoints fixed
• The fundamental group of a topological space $X$ at a basepoint $x_0$, denoted $\pi_1(X, x_0)$, is the set of path homotopy equivalence classes of loops based at $x_0$
• The group operation in the fundamental group is concatenation of loops
  • The identity element is the constant loop at the basepoint
  • The inverse of a loop is the same loop traversed in the opposite direction
• Knot groups are fundamental groups of knot complements in the 3-sphere $S^3$
• The Wirtinger presentation is a method for presenting knot groups using generators and relations derived from a knot diagram

Historical Context and Development

• The concept of the fundamental group was introduced by Henri Poincaré in the late 19th century
  • Poincaré's work laid the foundation for algebraic topology, which studies topological spaces using algebraic structures
• The fundamental group was initially used to study the topology of surfaces and higher-dimensional manifolds
• In the early 20th century, J.W. Alexander and Kurt Reidemeister developed the theory of knots and links
  • They recognized the importance of the fundamental group in distinguishing knots and links
• Ralph Fox introduced the concept of the knot group in the 1950s as a tool for studying knots
• The Wirtinger presentation of knot groups was developed by Wilhelm Wirtinger in the early 20th century
  • This presentation provides a systematic way to compute knot groups from knot diagrams
• The study of knot groups has led to significant advances in low-dimensional topology and geometric group theory
• The fundamental group and knot groups continue to be active areas of research in modern mathematics

The Fundamental Group: Basics

• The fundamental group is a group associated with a topological space that captures information about the space's loops and paths
• To define the fundamental group, we first choose a basepoint $x_0$ in the topological space $X$
• Elements of the fundamental group are equivalence classes of loops based at $x_0$, where two loops are equivalent if they are path homotopic
  • A loop is a continuous function $f: [0, 1] \to X$ with $f(0) = f(1) = x_0$
• The group operation is defined by concatenating loops: given two loops $f$ and $g$, their product $f * g$ is the loop obtained by first traversing $f$ and then traversing $g$
• The constant loop at the basepoint serves as the identity element of the fundamental group
• The inverse of a loop is the same loop traversed in the opposite direction
• The fundamental group is independent of the choice of basepoint for path-connected spaces

Homotopy and Path Equivalence

• Homotopy is a central concept in algebraic topology that formalizes the idea of continuous deformation
• A homotopy between two continuous functions $f, g: X \to Y$ is a continuous function $H: X \times [0, 1] \to Y$ such that $H(x, 0) = f(x)$ and $H(x, 1) = g(x)$ for all $x \in X$
  • Intuitively, a homotopy is a continuous family of functions that "interpolates" between $f$ and $g$
• Two functions are said to be homotopic if there exists a homotopy between them
  • Homotopy is an equivalence relation on the set of continuous functions from $X$ to $Y$
• Path homotopy is a special case of homotopy where the functions are paths with fixed endpoints
• Two paths $\alpha, \beta: [0, 1] \to X$ with $\alpha(0) = \beta(0) = x_0$ and $\alpha(1) = \beta(1) = x_1$ are path homotopic if there exists a continuous function $H: [0, 1] \times [0, 1] \to X$ such that:
  • $H(s, 0) = \alpha(s)$ and $H(s, 1) = \beta(s)$ for all $s \in [0, 1]$
  • $H(0, t) = x_0$ and $H(1, t) = x_1$ for all $t \in [0, 1]$
• Path homotopy is an equivalence relation on the set of paths with fixed endpoints
  • The equivalence classes under this relation are the elements of the fundamental group

Calculating the Fundamental Group

• There are several methods for calculating the fundamental group of a topological space
• For simple spaces like the circle $S^1$, the fundamental group can be determined directly from the definition
  • The fundamental group of $S^1$ is isomorphic to the integers $\mathbb{Z}$ under addition
• The Seifert-van Kampen theorem is a powerful tool for computing fundamental groups of spaces that can be decomposed into simpler pieces
  • This theorem relates the fundamental group of a space to the fundamental groups of its constituent parts
• For CW complexes, the fundamental group can be calculated using the cellular approximation theorem
  • This theorem allows us to work with cellular maps and cellular homotopies, which are combinatorial in nature
• In some cases, the fundamental group can be determined by constructing a universal covering space and using lifting properties
  • The fundamental group is isomorphic to the group of deck transformations of the universal cover
• For knots and links, the knot group can be calculated using the Wirtinger presentation (discussed in a later section)
• Computational tools, such as the Reidemeister-Schreier method and the Todd-Coxeter algorithm, can be used to simplify presentations and perform calculations in the fundamental group

Introduction to Knot Groups

• Knot groups are fundamental groups of knot complements in the 3-sphere $S^3$
• Given a knot $K$ in $S^3$, the knot complement is the space $S^3 \setminus K$, obtained by removing the knot from the 3-sphere
• The knot group of $K$, denoted $G(K)$, is the fundamental group of the knot complement, $\pi_1(S^3 \setminus K)$
  • The basepoint for the fundamental group is chosen to be a point on the boundary of a tubular neighborhood of the knot
• Knot groups are important invariants that can distinguish between different knots
  • If two knots have non-isomorphic knot groups, then the knots are necessarily distinct
• The knot group captures information about the "knotting" of the knot and the topology of the knot complement
• Knot groups have been used to study properties of knots, such as chirality, invertibility, and concordance
• The abelianization of the knot group is always isomorphic to the integers $\mathbb{Z}$, regardless of the specific knot
  • This fact is a consequence of Alexander duality and the homology of knot complements

Wirtinger Presentation of Knot Groups

• The Wirtinger presentation is a method for presenting knot groups using generators and relations derived from a knot diagram
• To construct the Wirtinger presentation, first choose an oriented knot diagram for the knot $K$
• Assign a generator $x_i$ to each arc (segment between crossings) in the diagram
• At each crossing, introduce a relation based on the orientation of the strands
  • If the strands are oriented as in the figure, the relation is $x_k = x_i^{-1} x_j x_i$, where $x_i$ corresponds to the overcrossing strand
• The Wirtinger presentation of the knot group $G(K)$ has generators $\{x_1, \ldots, x_n\}$ and relations determined by the crossings
• The Wirtinger presentation provides a systematic way to compute knot groups directly from knot diagrams
• Different diagrams of the same knot will yield different Wirtinger presentations, but the resulting groups will be isomorphic
  • Reidemeister moves on the knot diagram correspond to Tietze transformations on the group presentation
• The Wirtinger presentation can be simplified using techniques from combinatorial group theory, such as free reduction and Tietze transformations

Applications and Examples

• The fundamental group is a powerful tool for studying the topology of spaces and has numerous applications across mathematics
• In algebraic topology, the fundamental group is used to classify covering spaces and to define higher homotopy groups
  • The theory of covering spaces is closely related to the theory of principal bundles and has applications in geometry and physics
• The fundamental group can detect holes and obstructions in a space
  • For example, the fundamental group of a circle is $\mathbb{Z}$, while the fundamental group of a disk is trivial
• Knot groups have been used to study the properties and classification of knots and links
  • The unknotting problem, which asks whether a given knot is equivalent to the trivial knot, can be approached using knot groups
  • The knot group can detect chirality (handedness) of knots, as chiral knots have non-isomorphic knot groups
• The fundamental group has applications in physics, particularly in the study of topological defects and phase transitions
  • Topological defects, such as vortices and monopoles, can be classified using the fundamental group of the order parameter space
• In robotics and motion planning, the fundamental group is used to study the configuration spaces of mechanical systems
  • The fundamental group can detect obstacles and characterize the connectivity of the configuration space

Connections to Other Areas of Mathematics

• The fundamental group is a central object in algebraic topology and has deep connections to other areas of mathematics
• Group theory: The fundamental group is an important example of a group, and its structure reflects the topology of the underlying space
  • Techniques from combinatorial and geometric group theory are used to study fundamental groups and their presentations
• Homology theory: The fundamental group is related to the first homology group of a space via the Hurewicz theorem
  • The abelianization of the fundamental group is isomorphic to the first homology group with integer coefficients
• Covering space theory: The theory of covering spaces is intimately connected to the fundamental group
  • The fundamental group acts on the universal covering space by deck transformations, and the structure of this action determines the covering spaces of the base space
• Knot theory: Knot groups are fundamental groups of knot complements and are essential tools in the study of knots and links
  • The peripheral subgroup of a knot group, generated by the meridian and longitude, encodes important information about the knot
• Low-dimensional topology: The fundamental group plays a crucial role in the classification of surfaces and 3-manifolds
  • The geometrization conjecture, proved by Perelman, relates the geometry of a 3-manifold to its fundamental group
• Algebraic geometry: The fundamental group of a complex algebraic variety is an important invariant that captures the topology of the variety
  • The study of fundamental groups in this context leads to the theory of étale fundamental groups and Galois groups
• Mathematical physics: The fundamental group appears in the study of gauge theories and topological field theories
  • The fundamental group of the configuration space of a physical system determines the possible topological charges and defects in the system