4.1 Introduction to the fundamental group

The fundamental group is a powerful tool in knot theory, helping us classify knots and links. It captures information about loops and paths in a space, remaining unchanged under continuous deformations. This makes it invaluable for distinguishing between different knots.

Computing the fundamental group involves choosing a basepoint and considering loops. For simple spaces, we can use properties like contractibility and product spaces. In knot theory, we focus on the fundamental group of a knot's complement to determine its uniqueness.

Fundamental Group and Knot Theory

Fundamental group of topological spaces

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• The fundamental group $\pi_1(X)$ is a group associated with a topological space $X$ that captures information about loops and paths in the space
• Elements of the fundamental group are equivalence classes of loops, where two loops are equivalent if they can be continuously deformed into each other (homotopic)
• The fundamental group is a topological invariant, meaning it remains unchanged under continuous deformations (homeomorphisms) of the space
• In knot theory, the fundamental group is used to study and classify knots and links by examining the fundamental group of the complement of a knot or link (the space obtained by removing the knot or link from the 3-dimensional space $\mathbb{R}^3$ or $S^3$)
• Knots and links with different fundamental groups of their complements are considered distinct

Computation of fundamental groups

• To compute the fundamental group, choose a basepoint $x_0$ in the space $X$ and consider loops based at $x_0$
• For simple spaces, the fundamental group can be computed using the following properties:
  • The fundamental group of a contractible space is trivial: $\pi_1(X) = \{e\}$ (point, disk)
  • The fundamental group of a circle $S^1$ is isomorphic to the integers: $\pi_1(S^1) \cong \mathbb{Z}$
  • If $X$ is the product of two spaces $X_1$ and $X_2$, then $\pi_1(X) \cong \pi_1(X_1) \times \pi_1(X_2)$
  • If $X$ is the wedge sum of two spaces $X_1$ and $X_2$, then $\pi_1(X) \cong \pi_1(X_1) * \pi_1(X_2)$ (free product of groups)
• The fundamental group of a torus $T = S^1 \times S^1$ is $\pi_1(T) \cong \mathbb{Z} \times \mathbb{Z}$
• The fundamental group of a figure-eight space (wedge sum of two circles) is $\pi_1(X) \cong \mathbb{Z} * \mathbb{Z}$ (free group on two generators)

Homotopy vs fundamental group

• Two continuous functions $f, g: X \to Y$ are homotopic if there exists 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 $x \in X$
• Homotopy is an equivalence relation on the set of continuous functions from $X$ to $Y$
• If two loops $\alpha$ and $\beta$ based at $x_0$ are homotopic, then they represent the same element in the fundamental group $\pi_1(X, x_0)$
• Homotopy invariance: If $f: X \to Y$ is a continuous function and $f_*: \pi_1(X, x_0) \to \pi_1(Y, f(x_0))$ is the induced homomorphism, then homotopic maps induce the same homomorphism on fundamental groups

Applications in knot classification

• The fundamental group of the complement of a knot or link can be used to distinguish between different knots and links
• The unknot (trivial knot) has a complement with fundamental group isomorphic to $\mathbb{Z}$
• The trefoil knot has a complement with fundamental group presented by $\langle a, b | a^3 = b^2 \rangle$
• The Hopf link has a complement with fundamental group presented by $\langle a, b | [a, b] = 1 \rangle$ (free abelian group of rank 2)
• If two knots or links have complements with non-isomorphic fundamental groups, then the knots or links are distinct
• However, the converse is not true: there exist distinct knots and links with complements having isomorphic fundamental groups (square knot, granny knot)