Problem 7 - A First Course in Algebraic Topology

Question

\text { Show that torus knots of type }(\mathrm{m}, 1) \text { or }(1, n) \text { can be unknotted. }

Step-by-Step Solution

Answer

Torus knots of type \((m, 1)\) or \((1, n)\) can be unknotted because they can be deformed into a simple loop.

Step 1: Understanding Torus Knots

Torus knots are defined as knots that lie on the surface of a torus and can be characterized by two coprime integers \((m, n)\). The torus knot \((m, n)\) winds around the torus \(m\) times longitudinally and \(n\) times meridionally.

Step 2: Torus Knot of Type (m, 1)

Consider a torus knot of type \((m, 1)\). This implies that the knot winds around the torus \(m\) times longitudinally and crosses the meridian only once. Since it intersects the meridian once, the knot can be continuously deformed into a single cycle, making it equivalent to the unknot, which is a simple loop.

Step 3: Torus Knot of Type (1, n)

Similarly, a torus knot of type \((1, n)\) implies that the knot winds around the torus once longitudinally and \(n\) times meridionally. Due to the single longitudinal winding, the knot can be easily deformed into a simple loop, which again is equivalent to the unknot.

Key Concepts

Topology

Topology is a fascinating field within mathematics that focuses on the properties of space that are preserved under continuous transformations. It provides a way to consider shapes and surfaces that might look different but can be transformed into one another without cutting or tearing.
For example, a coffee cup and a donut (or torus) are considered the same in topology because they can be continuously deformed into each other without any breaks.

In dealing with torus knots, topology helps us understand how these knots can be manipulated or "untangled" from the torus surface. The key idea is to visualize how a knot may be adjusted or deformed smoothly across the surface, while keeping its inherent properties intact. Topology provides a lens to determine whether a complex knot can be turned into a simpler form, such as the unknot.

Knot Theory

Knot theory is a branch of topology dedicated exclusively to the study of knots. While in everyday life, a knot might mean tying up your shoes or securing a rope, in mathematics, a knot is a closed loop in a 3-dimensional space that doesn't intersect itself.
Knot theory aims to classify these knots and understand their properties. Each knot can be studied based on its unique structure, which is described by its crossing patterns and the ways it can be manipulated.

In our context, torus knots - are special because they lie on a torus and - are categorized by two numbers \((m,n)\). These numbers tell us how many times the knot circles around the longitudinal and meridional axes of the torus. By analyzing these patterns, we gain insights into how specific knots, like those of type \((m, 1)\) or \((1, n)\), can be considered equivalent to simpler knots, such as the unknot.

Unknot

The unknot is an essential concept in knot theory as it represents the simplest form of a knot—a perfect loop or circle with no crossings. When a complex knot can be smoothly adjusted into this simple loop, it is said to be equivalent to the unknot.
The significance of the unknot lies in our ability to determine the complexity of different knots. If a knot can be transformed into an unknot, it indicates that despite its appearance, the knot is not truly "complicated" in a topological sense.

Applying this to torus knots, we see that specific torus knots of types \((m, 1)\) or \((1, n)\) can indeed be deformed into an unknot. This implies that although they initially appear tangled while wound around the torus, they share the simple topology of a loop and do not possess any intrinsic knottedness when properly adjusted.