Problem 9 - A First Course in Algebraic Topology

Question

Torus knots were defined for positive pairs of coprime integers \((\mathrm{m}, \mathrm{n})\), but it makes sense to define torus knots for any pair of coprime (non-zero) integers. Show that if we change the sign of \(\mathrm{m}\) and/or of \(\mathrm{n}\) then the resulting torus knot is similar to the original one. Is it equivalent to the original one?

Step-by-Step Solution

Answer

The torus knot is similar, but not equivalent to the original when the sign of \(m\) or \(n\) is changed.

Step 1: Understanding Torus Knots

A torus knot is a type of knot that lies on the surface of a torus (a doughnut-shaped surface). It is characterized by two coprime integers \(m\) and \(n\). The pair specifies the number of times the knot winds around the torus in the longitudinal and meridional directions.

Step 2: Investigating Knot Similarity and Equivalence

Knot similarity involves geometric or topological aspects where the knot can be smoothly deformed without cutting or gluing, whereas equivalence requires the knot to be isotopic, meaning that it can be transformed into the other via a continuous deformation.

Step 3: Changing Signs and Its Effect

Consider changing the sign of \(m\) or \(n\). Let's first change the sign of \(m\): if \((m, n)\) becomes \((-m, n)\), the path of the knot on the torus is reversed in one direction. Similarly, changing the sign of \(n\) makes \((m, n)\) into \((m, -n)\), reversing the path in the other direction.

Step 4: Analyzing Similarity

Regardless of the signs, the trace of the knot remains on the same torus. The change only reflects as a different orientation. The knot still participant in the same topological space, hence is similar to the original in the broader geometric sense.

Step 5: Evaluating Equivalence

Although similar, torus knots with different sign pairs like \((m, n)\), \((-m, n)\), \((m, -n)\), and \((-m, -n)\) wind around the torus in opposite senses. Therefore, they are not equivalent in the sense of knot isotopy, as they cannot be continuously transformed into each other without cutting.

Key Concepts

Coprime Integers

Coprime integers are a fundamental aspect of defining torus knots. Two numbers are considered coprime if they do not have any common positive integer divisors other than 1. In the context of a torus knot, the pair \(m, n\) is used to describe how many times the knot winds around the torus' two principal cycles. It's important that \(m\) and \(n\) are coprime, because this condition ensures a simple and non-repeating pattern for the knot, giving it its complex and intriguing structure on the torus surface.

• A coprime pair ensures the knot doesn't trace the same path repeatedly.
• This unique path gives the torus knot its distinctive topology.
• If \(m\) and \(n\) were not coprime, the knot would not close up neatly and repeat its previous steps.

Understanding the concept of coprime numbers helps to grasp why torus knots are formed with such a specific and fascinating structure.

Knot Similarity

Knot similarity involves comparing two knots to see if they are similar in shape and structure without being necessarily identical. For torus knots, similarity can be visualized by considering how the knot is drawn on the torus without any strict requirement on the direction. When you change the sign of one of the pair of integers \(m\) or \(n\), the resulting torus knot still lies smoothly on the torus surface, maintaining its geometric integrity.

• Similarity allows small deformations but keeps the overall topology intact.
• Even if one direction of winding is reversed, the fundamental loop pathway remains unchanged.
• This broader view of similarity means that the basic shape and twist of the knot are conserved despite sign changes.

The concept of similarity provides insight into how flexible yet uniquely defined torus knots can be.

Knot Equivalence

Knot equivalence is more stringent compared to similarity, as it requires two knots to be isotopic to each other. This means you should be able to morph one knot into the other smoothly through space without any cuts or breaks. In terms of torus knots defined by \(m\) and \(n\), determining equivalence involves checking if the knots can seamlessly transition into each other.

• Equivalent knots must not only look similar but also must "feel" identical in their spatial presence.
• Sign changes in \(m\) or \(n\) create knots that don’t transition smoothly between each other.
• As such, torus knots with reversed sign directions are not equivalent under stricter isotopic transformations.

Understanding the nuanced distinction between equivalence and similarity can lead to deeper insights into the subtle properties of knots.

Orientation

Orientation in the context of knots refers to the direction in which the knot is tied around the underlying surface, such as the torus. The direction matters because it affects the knot’s properties, particularly when comparing knots for equivalence or similarity. When either \(m\) or \(n\) or both are given a negative sign, it changes the orientation of the torus knot.

• A change in orientation equates to a reversal of direction in one or both cycles on the torus.
• Oriented differently, a knot might maintain a visual similarity, but its properties differ fundamentally.
• This different orientation is a key reason why knots with reversed sign pairs are not equivalent.

Orientation is central to understating why such simple numerical changes (sign changes) can have such a profound impact on the knot.

Isotopy

Isotopy in knots means transforming one knot into another through continuous deformations without any cuts. This concept is at the heart of verifying whether two knots are equivalent. For torus knots defined by the integer pairs \(m, n\), isotopy checks if sign changes in either \(m\) or \(n\) allow for a seamless continuous transition.

• If a knot can be twisted, stretched, or otherwise deformed into another without tearing, they are deemed isotopic.
• Reversing signs in \(m\) or \(n\) alters the direction so drastically that knots become non-isotopic.
• Thus, while knots might appear similar after a sign change, they cannot be smoothly morphed into each other.

Isotopy underscores the physical and spatial properties that define knot equivalence and helps to separate truly equivalent knots from merely similar ones.