Question

Below is a common outline of a proof that the trefoil knot is nontrivial (i.e., not isotopic to the unknot). There are various ways to prove this, but two standard methods involve (1) using the fundamental group of the knot complement, or (2) using the Alexander polynomial. We sketch both approaches.

────────────────────────────────────────────────────────── 1. PROOF VIA THE FUNDAMENTAL GROUP OF THE KNOT COMPLEMENT ──────────────────────────────────────────────────────────

1.1. Definition of the Knot Group
Given a knot K in S³ (the 3-sphere, or equivalently R³ plus a point at infinity), let X = S³ − N(K) be the complement of a small open tubular neighborhood N(K) of K. Then π₁(X), the fundamental group of X, is called the knot group of K.

1.2. Knot Group of the Unknot
For the unknot U, its complement Xᵤ can be deformation retracted onto a circle surrounding the knot. This circle is a meridian, and the fundamental group is isomorphic to ℤ. Concretely,
π₁(S³ − N(U)) ≅ ℤ.

1.3. Knot Group of the Trefoil
One can show (using Wirtinger presentations, for example) that the trefoil knot T has a knot group π₁(S³ − N(T)) with presentation of the form
⟨ x, y ∣ x y x = y x y ⟩.

A more standard form is
⟨ x, y ∣ x⁻¹y⁻¹xy = y⁻¹x⁻¹yx ⟩,
but all these presentations are isomorphic. Crucially, this group is not isomorphic to ℤ.

1.4. Concluding Nontriviality
Since the trefoil’s group π₁(S³ − N(T)) is different from the trivial knot’s group π₁(S³ − N(U)) ≅ ℤ, these two complements are not homeomorphic. Hence the trefoil cannot be “un-knotted” by a continuous deformation in 3D: it is a nontrivial knot.

──────────────────────────────────────── 2. PROOF VIA THE ALEXANDER POLYNOMIAL ────────────────────────────────────────

2.1. Alexander Polynomial Overview
For any (tame) knot K, one can define a polynomial Δₖ(t), called the Alexander polynomial, which is a topological invariant of the knot. Informally, if you have two knots K₁ and K₂ that are isotopic, they have the same Alexander polynomial. Conversely, in particular, if K is the unknot, its Alexander polynomial is 1.

2.2. Alexander Polynomial of the Unknot
One can compute or look up that for the unknot U,
Δᵤ(t) = 1.

2.3. Alexander Polynomial of the Trefoil
A standard calculation or reference shows that the trefoil T has Alexander polynomial
Δₜ(t) = t² − t + 1.

2.4. Comparing to the Unknot
Since t² − t + 1 ≠ 1 as a polynomial in t, the trefoil’s Alexander polynomial differs from that of the unknot. Thus the trefoil cannot be isotopic to the unknot.

──────────────────────────────────────── 3. KEY TAKEAWAYS ────────────────────────────────────────

• Fundamental Group Method: The group π₁(S³ − N(K)) is ℤ for the unknot and is non-abelian for the trefoil, so they differ.
• Alexander Polynomial Method: The trefoil’s Alexander polynomial is t² - t + 1, not equal to 1, which is the Alexander polynomial for the unknot.

Each of these invariants (knot group or Alexander polynomial) detects that the trefoil is genuinely knotted and cannot be deformed to an unlinked circle in 3D space. Hence the trefoil is a nontrivial knot.

Okay, so I need to prove that the trefoil knot is actually knotted. Hmm, let me start by recalling what a trefoil knot is. From what I remember, the trefoil is the simplest non-trivial knot, which means it's not just the unknot or a circle. But how do I formally prove that it's knotted? Maybe I can use some knot invariants.

First, let me think about the fundamental group of the knot complement. If two knots have different fundamental groups, then they are not equivalent. The trefoil is a torus knot, right? So its fundamental group should be different from the fundamental group of the unknot complement, which is just the free group on one generator. Wait, the complement of the unknot in S^3 is a solid torus, so its fundamental group is Z. But the trefoil's fundamental group is non-abelian, isn't it? That would show they're different.

But maybe I need to compute the fundamental group of the trefoil complement. I think the Wirtinger presentation can be used here. Let me try to recall how that works. For a knot diagram, you assign a generator to each arc and a relation at each crossing. For the trefoil, which has three crossings, so three arcs and three relations. Let me draw a mental picture: the standard trefoil has three crossings, each with the overhand going over one strand.

So, labeling the arcs as a, b, c in order around the knot. At each crossing, the relation would be something like the generator of the overcrossing conjugating the generator of the undercrossing. Wait, the Wirtinger relations are of the form: if you have a crossing where arc x goes over arcs y and z, then the relation is z = x y x^{-1} or something like that. Maybe I need to be more precise.

Alternatively, maybe I can use the fact that the trefoil is a (2,3)-torus knot. The fundamental group of the complement of a (p,q)-torus knot has a presentation with two generators a and b and the relation a^p = b^q. For the trefoil, that would be a^2 = b^3. Then, the fundamental group is ⟨ a, b | a^2 = b^3 ⟩. This group is non-abelian because if you consider the elements a and b, a^2 = b^3 is in the center, but the group itself isn't abelian. For example, a and b don't commute. So since the fundamental group of the trefoil complement is non-abelian and the unknot complement has fundamental group Z, which is abelian, they can't be isomorphic. Therefore, the trefoil is not the unknot.

But wait, maybe I should check if the trefoil's group is actually non-abelian. Let me see. Suppose in the group ⟨ a, b | a^2 = b^3 ⟩, does ab = ba? If they did, then the group would be abelian, but let's see. If ab = ba, then a^2 b = a b a = b a a = b a^2. But a^2 = b^3, so substituting, we get b^3 b = b a^2 => b^4 = b a^2. But a^2 = b^3, so b^4 = b b^3 = b^4, which is a tautology. Hmm, that doesn't lead to a contradiction. Maybe another approach.

Alternatively, map this group to a non-abelian group. For example, map a to (12) and b to (123) in the symmetric group S3. Then a^2 is the identity, and b^3 is also the identity, so the relation a^2 = b^3 is satisfied. But ab = (12)(123) = (13), and ba = (123)(12) = (23), which are not equal. Therefore, the images of a and b don't commute, so the original group is non-abelian. Therefore, the fundamental group of the trefoil complement is non-abelian, while the unknot's is abelian, so they are different. Hence, the trefoil is knotted.

Alternatively, maybe use the Alexander polynomial. The Alexander polynomial of the unknot is 1, and the Alexander polynomial of the trefoil is t^2 - t + 1. Since they are different, the trefoil is not the unknot.

But wait, does having different Alexander polynomials necessarily mean the knots are different? I think yes, because the Alexander polynomial is a knot invariant. So if two knots have different Alexander polynomials, they are not equivalent. Since the trefoil has a non-trivial Alexander polynomial, it's knotted.

But maybe I need to compute the Alexander polynomial for the trefoil to be sure. Let me try. Using the Wirtinger presentation, the Alexander polynomial can be computed from the determinant of a certain matrix. For the trefoil, with three crossings, the Alexander matrix would be a 3x3 matrix. But maybe there's a simpler way.

Alternatively, the trefoil is a fibered knot with genus 1, and its Alexander polynomial is t^2 - t + 1. Since it's not 1, it's not the unknot.

Another approach: the trefoil has tricolorability. Wait, the unknot isn't tricolorable, but the trefoil is. So if a knot is tricolorable, it's non-trivial. But is tricolorability a sufficient condition? Yes, because the unknot can't be tricolored. Let me recall: tricoloring means you can color the strands with three colors such that at each crossing, either all three colors are the same or all three are different. The trefoil can be colored in a non-trivial way, alternating the colors, which satisfies the condition. Therefore, since it's tricolorable, it's knotted.

But maybe I need to check that tricolorability is preserved under Reidemeister moves. If so, then it's a knot invariant. Since the unknot isn't tricolorable, but the trefoil is, they can't be equivalent.

Alternatively, use the Jones polynomial. The Jones polynomial of the unknot is 1, and the Jones polynomial of the trefoil is t + t^3 - t^4. Since they are different, the trefoil is knotted.

But again, the question is to prove that the trefoil is knotted. So maybe the simplest way is to use one of these invariants. The fundamental group is a bit involved, but maybe the tricolorability is the easiest. Let me elaborate on that.

Tricolorability is an invariant of knots. A knot is tricolorable if each strand can be colored with one of three colors such that at each crossing, either all three colors are the same or all three are different. The unknot is not tricolorable because any attempt to color it would require only one color, but the trefoil can be colored with three colors. For example, take the standard diagram of the trefoil with three crossings. Assign each of the three arcs a different color. At each crossing, all three colors meet, so the condition is satisfied. Therefore, the trefoil is tricolorable, hence non-trivial.

But wait, actually, when you tricolor the trefoil, you can use three different colors, but you can also use just two colors if you alternate them. Wait, no, with three crossings, each arc is separate. Let me visualize: in the standard trefoil diagram, there are three arcs. If I color them red, blue, green in order, then at each crossing, all three colors meet. So each crossing has one of each color, satisfying the tricolor condition. Therefore, the trefoil is tricolorable.

On the other hand, the unknot cannot be tricolored. Suppose you try to tricolor the unknot: since there's only one arc, you have to color it all one color. Then, at each crossing (but there are no crossings in the standard unknot diagram), but if you consider a diagram with kinks, after removing the kinks via Reidemeister moves, you still end up with one color. Therefore, the unknot is not tricolorable.

Since tricolorability is preserved under Reidemeister moves, and the trefoil is tricolorable while the unknot is not, they cannot be equivalent knots. Therefore, the trefoil is knotted.

Alternatively, another approach is using the knot group's abelianization. The abelianization of the fundamental group of a knot complement is always Z, since the first homology group is Z. But the abelianization of the trefoil group is Z, same as the unknot. So that doesn't help.

But the key point is that the fundamental group of the trefoil is non-abelian, while the unknot's is abelian. So if we can show that the trefoil group is non-abelian, then they are different.

Earlier, I thought about mapping the trefoil group to S3, which is non-abelian. Since there's a non-abelian representation, the group itself must be non-abelian. Therefore, the trefoil's group is non-abelian, hence not isomorphic to Z, so the trefoil is knotted.

Alternatively, use the fact that the trefoil has genus 1. The genus of the unknot is 0, and genus is a knot invariant. Therefore, since the trefoil has genus 1, it's not the unknot.

But how do we know the genus of the trefoil is 1? Well, the trefoil is a (2,3)-torus knot, and the genus of a (p,q)-torus knot is (p-1)(q-1)/2. So (2-1)(3-1)/2 = 1*2/2 = 1. Therefore, genus 1. Since the unknot has genus 0, they are different.

But maybe proving the genus is 1 requires more work. Alternatively, using Seifert's algorithm to find a Seifert surface. For the trefoil, applying Seifert's algorithm to the standard diagram gives a surface with two disks connected by three twisted bands, which has genus 1. Therefore, the genus is 1, which is different from the unknot's genus 0.

So, multiple ways to show it.

Another thought: the trefoil is alternating. The unknot is also alternating, but the trefoil is a non-trivial alternating knot. However, just being alternating doesn't mean it's knotted. But maybe using properties of alternating knots. For example, an alternating knot is trivial if and only if it has a crossing number of 0, which is not the case here. The trefoil has crossing number 3, which is the minimum number of crossings needed to depict it. Since the unknot can be drawn with zero crossings (a circle), but any diagram of the unknot can be simplified to have no crossings, while the trefoil cannot be simplified below three crossings. Therefore, they are different.

But how do we know the trefoil cannot be unknotted? That's essentially what we're trying to prove. So maybe that's circular.

Alternatively, use the fact that the trefoil is a prime knot. The unknot is not considered a prime knot. But I think prime knots are non-trivial by definition.

Wait, maybe not. The unknot is a unit in the knot concordance group, but prime knots are those that cannot be expressed as a connected sum of two non-trivial knots. But the unknot is trivial, so maybe the trefoil is prime and non-trivial.

But this might not help directly.

Alternatively, use the hyperbolic volume. The trefoil knot is not hyperbolic, though; it's a torus knot, so its complement is Seifert fibered. The unknot complement is also not hyperbolic. So that might not help.

Wait, the complement of the trefoil is a Seifert fibered space with base orbifold a disk with two cone points of order 2 and 3. The unknot complement is a solid torus, which is also a Seifert fibered space with no cone points. Since their Seifert invariants are different, they are not homeomorphic. Therefore, the trefoil is knotted.

But this might be more advanced.

So, in summary, there are several ways to prove the trefoil is knotted:

1. Fundamental group of the complement is non-abelian, while the unknot's is abelian.

2. The Alexander polynomial of the trefoil is non-trivial.

3. The Jones polynomial is different.

4. The trefoil is tricolorable, while the unknot is not.

5. The genus of the trefoil is 1, while the unknot's is 0.

Each of these invariants distinguishes the trefoil from the unknot, thereby proving it's knotted.

But the problem says "prove that the trefoil is knotted," so maybe the expected answer is one of these proofs. Since the user might be looking for a concise proof, perhaps using tricolorability is the simplest.

Alternatively, using the fundamental group. Let me try to write that out more formally.

Consider the trefoil knot T. The fundamental group of its complement, π₁(S³ \ T), has a presentation ⟨ a, b | a² = b³ ⟩. This group is non-abelian, as shown by mapping a and b to non-commuting elements in S₃. Since the fundamental group of the unknot complement is ℤ, which is abelian, the groups cannot be isomorphic. Therefore, T is not the unknot, so it is knotted.

Alternatively, using tricolorability:

A knot is tricolorable if each arc can be colored with one of three colors such that at each crossing, either all three colors are present or all are the same. The trefoil knot can be tricolored by assigning each of its three arcs a distinct color. Since the unknot cannot be tricolored (as it has only one arc), tricolorability is an invariant that distinguishes them. Therefore, the trefoil is knotted.

Either of these would suffice, but maybe the fundamental group is more foundational, whereas tricolorability is a simpler combinatorial argument.

Given that the question is to prove the trefoil is knotted, and not necessarily to compute a specific invariant, perhaps the tricolorability argument is the most straightforward.

So, to recap:

1. Define tricolorability as a knot invariant.

2. Show that the trefoil can be tricolored.

3. Note that the unknot cannot be tricolored.

4. Conclude that the trefoil is not the unknot, hence knotted.

Yes, that seems efficient.

Alternatively, if I want to use the fundamental group, I need to compute it for the trefoil and show it's non-abelian, which is a bit more involved but also valid.

Given the problem is likely expecting a proof using a basic invariant, tricolorability is probably the way to go.

Final Answer The trefoil knot is non-trivially knotted, as demonstrated by its tricolorability—a property absent in the unknot. Thus, the trefoil cannot be equivalent to the unknot, proving it is knotted. \boxed{\text{Trefoil is knotted}}

To prove that the trefoil knot is knotted, we can use the concept of tricolorability, which is a knot invariant.

1. Tricolorability Definition: A knot is tricolorable if each strand can be colored with one of three colors such that at each crossing, either all three colors are the same or all three are different.

2. Tricolorability of the Trefoil Knot: The trefoil knot can be tricolored by assigning each of its three arcs a distinct color. In the standard diagram of the trefoil knot with three crossings, each crossing will have one of each color, satisfying the tricolor condition.

3. Tricolorability of the Unknot: The unknot cannot be tricolored. Any attempt to tricolor the unknot results in a single color, as it has only one arc and no crossings that meet the tricolor condition.

4. Conclusion: Since tricolorability is preserved under Reidemeister moves and the trefoil knot is tricolorable while the unknot is not, they cannot be equivalent. Therefore, the trefoil knot is knotted.

[ \boxed{\text{Trefoil is knotted}} ]

Okay, let's prove that the trefoil knot is knotted, meaning it's not equivalent to the unknot (a simple, unknotted loop). We'll do this by showing that a knot invariant, the tricolorability, distinguishes the trefoil from the unknot.

1. Knot Diagrams and Reidemeister Moves

• Knot Diagrams: We represent knots using knot diagrams, which are 2D projections of the knot onto a plane. Crossings in the diagram indicate where one part of the knot passes over another.

• Reidemeister Moves: Two knot diagrams represent the same knot (i.e., the knots are equivalent, or "ambient isotopic") if and only if you can transform one diagram into the other using a sequence of Reidemeister moves. These are local changes to the diagram that don't change the underlying topology of the knot. There are three types:

  • Type I (Twist/Untwist): Adding or removing a simple twist.
  • Type II (Poke/Slide): Sliding one strand over or under another.
  • Type III (Slide): Sliding a strand over or under a crossing.
  • Knot Invariants A knot invariant is a quantity or property associated with a knot which does not change after applying Reidemeister moves. Thus, if an invariant differs for two knot diagrams, they must represent different knots.

2. Tricolorability

Tricolorability is a knot invariant. A knot diagram is tricolorable if we can color the strands of the diagram using three distinct colors (let's say Red, Green, and Blue) subject to these rules:

• At least two colors must be used. (The entire knot cannot be the same color.)
• At each crossing, either all three strands have the same color, or all three strands have different colors. This is the crucial rule.

3. The Unknot is NOT Tricolorable

Consider the standard diagram of the unknot, a simple circle:

    O

Since there are no crossings, we have only one strand. We can color it with one color (say, Red). However, to be tricolorable, we must use at least two colors. Therefore, the unknot is not tricolorable.

4. The Trefoil IS Tricolorable

Now, consider the standard diagram of the trefoil knot:

       /\
      /  \
     /    \
    /______\
    \      /
     \    /
      \  /
       \/

There are three crossing in the diagram.

We can color it as follows:

• Give the top left crossing all red strands.
• Give the top right crossing all green strands.
• Then, color the bottom strands such that at each crossing, all three strands are different colors.

Here is a color assignment:

• The arc that goes from the top-left crossing, over the top-right crossing, and down to the bottom crossing: Red (at the top-left crossing), then Green (at the top-right crossing), then Blue (at the bottom crossing).
• The arc that goes from the top-right crossing, under the top-left crossing, and down to the bottom crossing: Green (at the top-right crossing), then Red (at the top-left crossing), then Blue (at the bottom crossing).
• The arc that goes from the bottom crossing, up and around to the top-left crossing: Blue, then Red.
• The remaining arc from the top left crossing to the top right, over the top: Red.

     R /\ R
      /  \
     G    G
    /______\
    B      B
     \    /
      \  /
     B \/ B

This is a valid tricoloring:

• At least two colors are used (in fact, all three are).
• At each crossing, either all three strands are the same color (top left all Red, top right all Green, bottom all Blue), or all three are different.

Thus, the trefoil knot is tricolorable.

5. The Proof

• The unknot is not tricolorable.
• The trefoil knot is tricolorable.
• Tricolorability is a knot invariant (it's preserved under Reidemeister moves. This needs to be shown separately, but it is a standard result. You prove it by exhaustively checking that none of the Reidemeister moves can change tricolorability).
• Since the trefoil and the unknot have different values for a knot invariant, they cannot be equivalent knots.

Therefore, the trefoil knot is knotted (i.e., it is not the unknot).

In Summary

We used the knot invariant of tricolorability to distinguish the trefoil from the unknot. Because the trefoil can be tricolored and the unknot cannot, and tricolorability is preserved by the Reidemeister moves, the trefoil cannot be deformed into the unknot, proving it is indeed knotted. This is a classic example of using a simple knot invariant to prove a fundamental property of knots.

Certainly! Let's delve into knot theory to prove that the trefoil knot is indeed a nontrivial (knotted) knot, meaning it cannot be transformed into an unknot (a simple loop) without cutting the strand.

1. Introduction to Knot Theory

• Knot: In mathematical terms, a knot is a closed, non-self-intersecting curve embedded in three-dimensional space, ( \mathbb{R}^3 ).
• Unknot: The simplest knot, which is equivalent to a standard loop or circle. It's considered "trivial" because it lacks any actual knotting.
• Trefoil Knot: The simplest nontrivial knot with three crossings. It resembles a three-leaf clover.

2. Objective

Prove that the trefoil knot is not equivalent to the unknot, i.e., it is a nontrivial knot.

3. Strategy for Proof

To establish that the trefoil is knotted (nontrivial), we'll utilize knot invariants—properties that remain unchanged under ambient isotopies (deformations that don't involve cutting the knot). If an invariant distinguishes the trefoil from the unknot, it proves that the trefoil is a nontrivial knot.

One effective invariant for this purpose is the Alexander Polynomial.

4. The Alexander Polynomial

• Definition: For a given knot, the Alexander polynomial ( \Delta(t) ) is a Laurent polynomial invariant under ambient isotopy.
• Property: The Alexander polynomial of the unknot is ( \Delta_{\text{unknot}}(t) = 1 ).

If the Alexander polynomial of a knot differs from 1, the knot is nontrivial.

5. Computing the Alexander Polynomial for the Trefoil Knot

There are multiple methods to compute the Alexander polynomial, such as using a Seifert surface or the knot's presentation. For simplicity, we'll cite the known result:

• Alexander Polynomial of the Trefoil: ( \Delta_{\text{trefoil}}(t) = t - 1 + t^{-1} ) or equivalently ( t^2 - t + 1 ) when normalized.

6. Comparing with the Unknot

• Unknot's Alexander Polynomial: ( \Delta_{\text{unknot}}(t) = 1 )
• Trefoil's Alexander Polynomial: ( \Delta_{\text{trefoil}}(t) = t^2 - t + 1 )

Since ( \Delta_{\text{trefoil}}(t) \neq \Delta_{\text{unknot}}(t) ), the trefoil knot is not equivalent to the unknot.

7. Alternative Invariant: Knot Group

Another way to demonstrate that the trefoil is knotted is by examining its fundamental group (also known as the knot group).

• Knot Group of the Unknot: Isomorphic to ( \mathbb{Z} ) (the integers under addition).
• Knot Group of the Trefoil: Presented as ( \langle a, b \ | \ a^2 = b^3 \rangle ), which is non-abelian (the group operation is not commutative).

Since the trefoil's knot group is not isomorphic to ( \mathbb{Z} ), it cannot be the unknot.

8. Conclusion

By evaluating the trefoil's Alexander polynomial and its knot group, we've established that:

• The Alexander polynomial of the trefoil differs from that of the unknot.
• The knot group of the trefoil is not isomorphic to that of the unknot.

Therefore, the trefoil knot is indeed a nontrivial (knotted) knot.

Further Reading

If you're interested in exploring more about knot invariants and knot theory, consider the following topics:

• Jones Polynomial: Another powerful knot invariant.
• Seifert Surfaces: Surfaces bounded by the knot which help in computing invariants.
• Knot Concordance and Knot Cobordism: Advanced topics in understanding knots in higher dimensions and their transformations.

Understanding these concepts provides deeper insights into the rich and fascinating world of knot theory.

To prove that the trefoil knot is indeed knotted, we need to show that it cannot be continuously deformed into a simple loop without passing through itself. Several methods can be used to demonstrate this, including the use of knot invariants, such as the Jones polynomial or the fundamental group of the knot complement.

Method 1: Using the Jones Polynomial

The Jones polynomial is a knot invariant that assigns a polynomial in the variable ( t^{1/2} ) to a knot. For the trefoil knot, the Jones polynomial is:

[ V_{\text{trefoil}}(t) = t + t^3 - t^4 ]

For the unknot (a simple loop), the Jones polynomial is:

[ V_{\text{unknot}}(t) = 1 ]

Since the Jones polynomials of the trefoil and the unknot are different, the trefoil cannot be deformed into the unknot without passing through itself. Therefore, the trefoil is knotted.

Method 2: Using the Fundamental Group

Another way to show that the trefoil is knotted is by examining the fundamental group of its complement in 3-dimensional space. The fundamental group of the trefoil knot complement is given by the presentation:

[ \pi_1(S^3 - \text{trefoil}) = \langle a, b \mid a^2 = b^3 \rangle ]

This group is isomorphic to the braid group on three strands, which is not abelian. In contrast, the fundamental group of the unknot complement is the integers, ( \mathbb{Z} ), which is abelian. Since the fundamental groups are not isomorphic, the trefoil knot cannot be deformed into the unknot, proving that it is knotted.

Method 3: Colorability

The trefoil knot is tricolorable, meaning it can be colored with three colors such that at each crossing, either all three colors meet or only one color is present. This is a property that the unknot does not have; the unknot is only singly colorable. Since tricolorability is a knot invariant, the difference in colorability indicates that the trefoil is not equivalent to the unknot.

Conclusion

Using any of these methods—Jones polynomial, fundamental group, or tricolorability—we can conclude that the trefoil knot is indeed knotted and cannot be continuously deformed into a simple loop without passing through itself. Therefore, the trefoil is a nontrivial knot. (\boxed{\text{The trefoil is knotted.}})

There are several classical proofs that the trefoil knot is nontrivial (i.e. “knotted”). One standard strategy is to show that some invariant that is preserved under ambient isotopy takes a different value for the trefoil than for the unknot. I’ll describe two proofs below.

────────────────────────────── Proof Using the Fundamental Group of the Knot Complement

1. In knot theory one associates to any knot K its complement in three‐space, X = ℝ³ \ K. It turns out that the fundamental group π₁(X) is an invariant of the embedding up to ambient isotopy.

2. For the unknot (i.e. a circle embedded in ℝ³ in the standard “round” way), one may show that its complement is homeomorphic to a solid torus whose fundamental group is isomorphic to ℤ (an abelian group).

3. For the trefoil knot, one may compute (using the Wirtinger presentation, for example) that the knot group has a presentation such as

π₁(Xₜ) ≅ ⟨a, b | a² = b³⟩.

This group is nonabelian. (Indeed, it is well known that this group is isomorphic to the braid group B₃ or to a 2–generator group that does not commute.)

1. Since the property “being abelian” is a group invariant and the fundamental group of the trefoil’s complement is nonabelian while that of the unknot is abelian, there is no homeomorphism between the exteriors of the trefoil and the unknot. In particular, they are not ambient isotopic, and so the trefoil is knotted.

────────────────────────────── Proof Using the Alexander Polynomial

1. The Alexander polynomial Δ_K(t) is another invariant of knots that is unchanged under ambient isotopy (up to multiplication by units ±tⁿ).

2. One can compute that the Alexander polynomial of the unknot is

Δ_unknot(t) = 1.

1. A computation (using a Seifert surface for the trefoil and “Seifert matrices” or directly via a presentation of the knot group) shows that the trefoil knot has

Δ_trefoil(t) = t − 1 + t⁻¹    (up to the usual normalization).

1. Since Δ_trefoil(t) ≠ 1, the trefoil cannot be ambient isotopic to the unknot. Thus the trefoil is knotted.

────────────────────────────── Conclusion

In either approach, one finds a topological invariant (the knot group or the Alexander polynomial) that distinguishes the trefoil from the unknot. Since these invariants are preserved under ambient isotopy, the trefoil cannot be deformed into the unknot and is therefore a nontrivial (knotted) knot.

This completes the proof that the trefoil is knotted.

To prove that the trefoil knot is indeed a non-trivial knot (i.e., it is knotted), we can demonstrate that it is not equivalent to the unknot (a simple loop) through knot invariants. One effective way to show this is by using the concept of the knot group or leveraging the Jones polynomial.

Using the Knot Group:

1. Definition: The knot group is the fundamental group of the complement of the knot in ( S^3 ). For a knot ( K ), this is defined as ( \pi_1(S^3 \setminus K) ).

2. Trefoil Knot Group: The trefoil knot can be represented in the plane with a diagram with three crossings. Its complement's fundamental group can be presented as: [ \pi_1(S^3 \setminus \text{Trefoil}) = \langle a, b \mid a^2 = b^3 \rangle ] This group is well-known and is isomorphic to the braid group ( B_3 ), which is non-abelian.

3. Knot Group of the Unknot: The knot group of the unknot (a simple loop) is ( \mathbb{Z} ) because its complement is simply connected.

4. Conclusion: Since the knot group of the trefoil is not isomorphic to the knot group of the unknot (as it is non-abelian compared to the abelian group (\mathbb{Z})), the trefoil is not equivalent to the unknot, thus it is knotted.

Using the Jones Polynomial:

1. Definition: The Jones polynomial ( V(t) ) is a polynomial invariant of an oriented knot, providing a powerful way to distinguish between different knots.

2. Trefoil Jones Polynomial: The Jones polynomial for the trefoil knot, which can be computed using the skein relation and some knot diagrams, is: [ V(t) = t + t^3 - t^4 ]

3. Jones Polynomial of the Unknot: The Jones polynomial for the unknot is ( V(t) = 1 ).

4. Conclusion: Since the Jones polynomial of the trefoil ( (t + t^3 - t^4) ) is different from that of the unknot ( (1) ), the trefoil knot cannot be the same as the unknot, confirming it is indeed a knotted structure.

Both methods show that the trefoil knot is not equivalent to the unknot, thereby proving it is knotted.

To prove that the trefoil knot is knotted, we need to show that it is not equivalent to the unknot (the circle). One way to do this is to use the concept of the tricolorability of a knot diagram.

Tricolorability:

A knot diagram is tricolorable if each arc of the diagram can be colored with one of three colors (say red, green, and blue) such that the following conditions hold:

1. At each crossing, either all three arcs have the same color, or all three arcs have different colors.
2. At least two colors are used.

Claim 1: The trefoil knot is tricolorable.

Consider the standard diagram of the trefoil knot. We can easily color it with three colors such that both conditions above are satisfied. You can pick any color for the first arc and work your way around, applying the rule at each crossing to determine the color of the remaining arcs. It's straightforward to find a tricoloring.

Claim 2: The unknot is not tricolorable.

Consider any diagram of the unknot. Because it is equivalent to a simple circle, we can draw it as a single arc with no crossings. If we want to color this, we can only use one color, which violates the second condition of tricolorability. If the diagram has crossings, then condition 1 requires that all arcs have the same color since there can be no mixed color crossings. Thus, only one color is used.

Claim 3: Tricolorability is an invariant.

This is the crucial step. If a knot diagram is tricolorable, then any other diagram representing the same knot is also tricolorable. In other words, the tricolorability of a knot is a property of the knot itself, not just of a particular diagram.

To prove this, we need to show that tricolorability is preserved under the Reidemeister moves. The Reidemeister moves are a set of three local moves that generate all possible equivalences between knot diagrams.

• Type I: Adding or removing a twist. You can check that if a diagram is tricolorable, performing a Type I move will result in a diagram that is also tricolorable.
• Type II: Adding or removing an overcrossing or undercrossing. Similarly, you can check that this move preserves tricolorability.
• Type III: Sliding a strand over or under a crossing. Again, this move preserves tricolorability.

Since tricolorability is preserved under the Reidemeister moves, it is an invariant of the knot.

Conclusion:

Since the trefoil knot is tricolorable and the unknot is not, and tricolorability is an invariant, the trefoil knot cannot be equivalent to the unknot. Therefore, the trefoil knot is knotted.

More Rigorous Proof (Using Knot Polynomials)

A more sophisticated proof uses knot polynomials. A knot polynomial is an invariant of a knot, meaning it is the same for any projection of the knot. The Alexander polynomial and the Jones polynomial are two common examples.

• The Alexander polynomial of the unknot is 1. The Alexander polynomial of the trefoil knot is t - 1 + t^-1. Since these polynomials are different, the unknot and trefoil knot are distinct.

• The Jones polynomial of the unknot is 1. The Jones polynomial of the trefoil knot is t + t³ - t⁴. Since these polynomials are different, the unknot and trefoil knot are distinct.

Using either of these methods provides a mathematically rigorous proof that the trefoil is knotted. The tricolorability proof, while less powerful, offers a more accessible introduction to the idea of knot invariants.