Problem 4 - A First Course in Algebraic Topology

Question

Find examples (if possible) of knots \(\mathrm{K}\) such that (i) \(\mathrm{K} \subset \mathrm{S}^{2} \subset \mathrm{R}^{3}\). (ii) \(\mathrm{K} \subset\) torus \(\subset \mathrm{R}^{3}\), (iii) \(\mathrm{K} \subset\) double torus \(\subset \mathrm{R}^{3}\).

Step-by-Step Solution

Answer

(i) Trefoil knot; (ii) (3,2)-torus knot; (iii) Composite knot made of torus knots.

Step 1: Understanding Knots

Knots are embeddings of a circle within three-dimensional space \(\mathbb{R}^3\). They can take various forms based on their enclosing surface.

Step 1: Knots within a sphere

A common occurrence of knots can be found within the three-dimensional sphere \(\mathrm{S}^2\). Essentially, any practical knot such as a trefoil knot can reside within a sphere. They take their form as a tangling of a closed loop which is not equivalent to an unknotted loop.

Step 2: Knots within a torus

Knots can also reside within a torus. One famous example is the torus knot itself. Within the donut-like structure of a torus, a (p, q)-torus knot wraps around the torus p times in one direction and q times in another, not returning to the initial position until these paths are completed.

Step 3: Knots within a double torus

A double torus (or genus 2 surface) is a surface with two "holes." It is more complex, allowing for a multi-component knot to exist within its structure, such as a sum or combination of multiple torus knots across the two toroidal loops.

Key Concepts

Sphere-Embedded Knots

Sphere-embedded knots are an interesting topic in topology. When we discuss knots within the context of a sphere, specifically the three-dimensional sphere \(\mathrm{S}^{2}\), it implies an embedding of the knot within a sphere's surface as part of \(\mathbb{R}^3\). In practical terms, imagine any knot that you could tie with a piece of string. You can conceptualize this knot as residing on the surface of a sphere.

- A classic example of a knot that fits this description is the trefoil knot. It is a simple type of knot that cannot be undone without cutting.
- The trefoil knot can be arranged in such a way that it remains within the confines of a sphere, serving as an excellent model of a sphere-embedded knot.

These types of knots offer a visually helpful way to understand the concept since nearly any knot you tie can theoretically be a sphere-embedded knot.

Torus Knot

Torus knots are special types of knots that lie on a torus, a surface that resembles the shape of a donut or inner tube. When a knot is said to be a torus knot, it must adhere to specific rules of behavior on this shape.

- A \(\text{(p, q)}\) torus knot is defined by its path around the torus: it winds around the central axis of the torus \(p\) times and through the hole \(q\) times before closing.
- These parameters, \(p\) and \(q\), must be coprime for the knot to be categorized as a torus knot.

Torus knots provide a fascinating study due to their combination of predictable behavior and complex appearance. They represent a wide range of potential configurations because of the varying values of \(p\) and \(q\), offering rich exploration possibilities in topology.

Genus 2 Surface Knot

Genus 2 surface knots are situated within a double torus, which is a more complex surface featuring two holes. The structure of a double torus allows for more intricate knot arrangements because there is more 'space' to work with.

- This topologically translates into multiple loops or components, giving rise to knots that can be more elaborate or involve multiple linked components.
- For example, the sum or split of torus knots can form fascinating knots on a double torus.

The increased complexity of genus 2 surface knots provides opportunities for exploration beyond simpler single-surface knots, broadening the understanding of how knots can behave in higher-genus surfaces.