A note on smoothly slice links in $S^2 \times S^2$

This paper provides an alternative proof that certain links are not smoothly slice in $S^2 \times S^2$ and explores the implications of this result for detecting exotic smooth structures on $S^2 \times S^2$.

The Gist

Imagine you are a master knot-tyer. You have a piece of string (a knot) or two pieces of string tied together (a link). In the world of mathematics, there's a special question: Can you untie this knot or link without cutting the string, if you are allowed to push it through a fourth dimension?

If you can push it through that extra dimension until it becomes a perfect, flat circle (or two separate circles) that can be shrunk down to nothing, mathematicians call it "slice."

This paper is about a specific, tricky puzzle: Can a specific two-loop knot be "sliced" if the fourth dimension is shaped like a donut made of two spheres stuck together (mathematicians call this $S^2 \times S^2$)?

Here is the story of how the authors, Marco and Clayton, solved this puzzle, explained with some everyday analogies.

1. The Setting: The "Donut" Universe

Most people know the 3D world we live in. But in math, we can imagine a 4D world.

• The Standard 4D World ($B^4$): Think of this as a giant, empty 4D room. If a knot can be untied here, it's "slice."
• The Special 4D World ($S^2 \times S^2$): This is like a 4D room that has a specific, bumpy shape. It's not empty; it has "holes" and "loops" built into its structure.
• The Goal: The authors wanted to find a knot that is easy to untie in the empty room but impossible to untie in this bumpy, special room.

2. The "Impossible" Knot

The authors constructed a specific two-loop knot (shown in Figure 1 of the paper). They didn't just guess; they built it like a Lego set using specific rules:

• It has two loops that twist around each other.
• The loops have specific "twistiness" (mathematicians call this the Arf invariant and signature).
• They are linked together in a very precise way (like two gears meshing).

3. The Detective Work: Ruling Out the Escape Routes

To prove this knot cannot be untied in the special 4D room, the authors played a game of "Elimination." They asked: "If this knot could be untied, what would the path look like?"

They imagined the two loops being pulled into the 4th dimension as two flat, smooth discs.

• The Problem: In this special 4D room, these discs can't just float anywhere. They have to fit into specific "slots" (homology classes).
• The Analogy: Imagine trying to park two cars in a very specific, weirdly shaped garage. The garage has rules about where cars can go based on their size and shape.

The authors used a series of "mathematical detectors" to check if the cars could park:

1. The "Shape" Detector (Genus Function):
   They checked the "size" of the knots. They knew that if the knots were untieable, the resulting discs would have to be a certain size. But the math showed that the required size didn't fit the shape of the garage. Result: Impossible.

2. The "Twist" Detector (Arf Invariant):
   They checked how much the knots were twisted. The garage has a rule: "If you park here, your twist must be even." But the knots had an odd twist. Result: Impossible.

3. The "Signature" Detector (Levine-Tristram Signatures):
   This is the most complex tool. Imagine the knot has a "sound signature" or a "frequency." When you try to push it into the 4D room, the room "resonates" at a certain frequency.
   The authors calculated the frequency of their knot. They found that if the knot were untieable, the room would have to vibrate at a frequency that simply doesn't exist in that specific 4D shape.

   • Analogy: It's like trying to fit a square peg into a round hole, but the hole is made of a material that screams if you try to force a square peg in. The math "screamed" (showed a contradiction).

4. The Big Reveal

By combining all these detectors, the authors proved that no matter how you try to push the knot into the 4th dimension, it gets stuck. It cannot be sliced.

This is a big deal because:

• It's a "Smooth" Proof: Previous mathematicians had shown this was impossible in a "rough" (topological) way, but this paper proves it's impossible even if you are allowed to wiggle and stretch the knot smoothly.
• The "Exotic" Possibility: The authors hint at a sci-fi twist. If you take this knot and perform a specific surgery (cutting and gluing) on it, you might create a universe that looks exactly like our standard $S^2 \times S^2$ from the outside, but is secretly different on the inside.
  • Analogy: Imagine two identical-looking houses. One is a normal house. The other is a "fake" house that looks the same but has a secret, twisted hallway inside that you can't see from the outside. This knot might be the key to building that "fake" house (called an Exotic 4-Manifold).

Summary

The authors built a specific, tricky two-loop knot. They used a battery of mathematical tests (like checking the knot's size, twist, and "frequency") to prove that this knot cannot be untied in a specific 4-dimensional shape called $S^2 \times S^2$.

This proves that not all knots are created equal in the 4th dimension. Furthermore, this discovery might be the first step toward finding "exotic" versions of 4D space—universes that look normal but feel completely different underneath.

Technical Summary

Here is a detailed technical summary of the paper "A Note on Smoothly Slice Links in $S^2 \times S^2$" by Marco Marengon and Clayton McDonald.

1. Problem Statement

The paper addresses the problem of smooth sliceness for links in the 4-manifold $X = S^2 \times S^2$.

• Definition: A link $L \subset S^3$ is smoothly slice in a 4-manifold $X$ if it bounds two disjoint, smoothly embedded discs in $X \setminus \text{Int}(B^4)$.
• Context: While it is a classical result that every knot is smoothly slice in $S^2 \times S^2$ (and in $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$), the status of multi-component links is more complex.
• Prior Work: Miyazaki and Yasuhara (1997) proved the existence of non-slice 2-component links in a punctured $S^2 \times S^2$ within the topological category. However, their methods rely on signature constraints (mod 8) that do not apply to the specific link constructed in this paper, and their topological proof cannot distinguish between standard and exotic smooth structures.
• Goal: The authors aim to construct a specific 2-component link that is not smoothly slice in $S^2 \times S^2$. Proving non-sliceness in the smooth category is a prerequisite for using such links to detect exotic $S^2 \times S^2$ manifolds (smooth manifolds homeomorphic but not diffeomorphic to the standard $S^2 \times S^2$).

2. Methodology

The authors employ a combination of obstruction theory, homological algebra, and knot invariants to rule out the existence of slice discs for a specific link $L = A \cup B$.

A. The Link Construction

The link $L$ is defined by a specific diagram (Figure 2) involving two $(1,1)$-tangles $T_A$ and $T_B$ and a region with $n$ right-handed full twists. The authors impose a series of assumptions (A1–A8) on the components $A$ and $B$:

• Symmetry: $A$ and $B$ are ambient isotopic (swappable).
• Genus: Both components have 4-ball genus $g_4 = 1$.
• Linking Number: $\text{lk}(A, B) = -4$.
• Arf Invariant: $\text{Arf}(A) = \text{Arf}(B) = 1$.
• Signatures: Specific values for the Levine-Tristram signature functions at roots of unity $\zeta_2, \zeta_4, \zeta_8$ (all equal to 2).
• Concrete Example: The authors identify $K = m(7_2)$ (the mirror of the knot $7_2$) as a knot satisfying these properties, setting$A=B=K$.

B. Obstructive Framework

The proof proceeds by contradiction. Assume $L$ bounds disjoint smooth discs $D_A, D_B$ in $X^\circ = (S^2 \times S^2) \setminus B^4$ with homology classes $\alpha, \beta \in H_2(S^2 \times S^2) \cong \mathbb{Z}^2$. The authors systematically eliminate all possible pairs $(\alpha, \beta)$ using three main tools:

1. Ruberman's Genus Function (Theorem 3.2):

   • This theorem calculates the minimal genus of a smooth surface in $S^2 \times S^2$ for a given homology class $(a_1, a_2)$.
   • The formula is $G(a_1, a_2) = 0$ if $a_1 a_2 = 0$, and $(|a_1|-1)(|a_2|-1)$ otherwise.
   • Application: By capping the slice discs with minimal genus surfaces in $B^4$, the authors construct closed surfaces in $S^2 \times S^2$. They show that for certain candidate homology classes, the required genus (derived from the knot properties) contradicts the minimal genus allowed by the homology class.

2. Arf Invariant Constraints (Theorem 2.3):

   • This theorem relates the signature of the manifold, the self-intersection of a characteristic surface, and the Arf invariant of the boundary knot.
   • Application: Used to rule out cases where the homology classes are characteristic (e.g., $\beta^2=0$) but the knot's Arf invariant is 1, creating a contradiction modulo 2.

3. Levine-Tristram Signatures (Theorem 2.1):

   • This provides an inequality bounding the signature of a knot based on the homology class of the surface it bounds.
   • Application: The authors use specific values of the signature function at roots of unity ($\zeta_2, \zeta_4, \zeta_8$) to derive inequalities. For example, if a knot bounds a disc in a class divisible by $m$, the signature must satisfy specific bounds. The authors choose signature values for $A$ and $B$ such that the sum of signatures for derived knots (like $A \# B$ or cables) violates these bounds.

3. Key Results and Proof Structure

The proof is a rigorous case analysis (Section 3) that eliminates all possible homology pairs $(\alpha, \beta)$:

1. Reduction via Symmetry: Using the symmetries of $S^2 \times S^2$ and the link, the authors reduce the infinite set of possible homology pairs to a finite list of "infinite families" and "sporadic cases" (Lemma 3.4).
2. Elimination of Families:
   • Family (1): Ruled out using the genus function (Lemma 3.5). The homology class $\alpha + \beta$ would require a genus-2 surface, but the minimal genus for that class is higher.
   • Family (2): Ruled out using Levine-Tristram signatures at $\zeta_2$ (Lemma 3.6). The signature of the connected sum $A \# B$ violates the bound derived from the homology class.
3. Elimination of Sporadic Cases:
   • Cases (5) and (6) are ruled out by the signature bound at $\zeta_2$ (Lemma 3.6).
   • Case (4) $((2,2), (-1,3))$ is ruled out by applying Theorem 2.1 with $m=8$ to a cable knot, using signatures at $\zeta_8$ (Lemma 3.7).
4. Conclusion: Since all possible homology classes for the slice discs lead to contradictions, the link cannot be smoothly slice.

Theorem 1.1: The specific 2-component link constructed (using $K=m(7_2)$) is not smoothly slice in $S^2 \times S^2$.

4. Significance and Applications

A. Distinction from Topological Results

The paper highlights a crucial distinction between the smooth and topological categories. While Miyazaki and Yasuhara proved non-sliceness in the topological category, their methods do not apply here. The authors' proof is strictly smooth, meaning the link might be topologically slice. This distinction is vital because smooth non-sliceness is the necessary condition for detecting exotic smooth structures.

B. Detection of Exotic $S^2 \times S^2$

The primary significance of this result lies in Proposition 4.1, which outlines a method to construct exotic 4-manifolds:

• If a link $L$ is not smoothly slice in the standard $S^2 \times S^2$, one can perform surgery on $L$ with specific framings to create a trace $X$.
• If this trace can be capped off by a rational homology ball $W$ to form a closed manifold $X' = X \cup -W$, and if $X'$ is simply connected and spin, then $X'$ is homeomorphic to $S^2 \times S^2$ (by Freedman's classification).
• However, because $L$ is not smoothly slice in the standard $S^2 \times S^2$ but is slice in $X'$ (by construction), $X'$ cannot be diffeomorphic to the standard $S^2 \times S^2$.
• Thus, $X'$ would be an exotic $S^2 \times S^2$.

This paper provides the first concrete candidate link for such a construction in $S^2 \times S^2$, analogous to their previous work on $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$.

Summary

Marengon and McDonald provide a rigorous proof that a specific 2-component link is not smoothly slice in $S^2 \times S^2$. By synthesizing the genus function, Arf invariants, and Levine-Tristram signatures, they eliminate all potential homological obstructions. This result is a foundational step toward the potential discovery of exotic smooth structures on $S^2 \times S^2$, a major open problem in 4-manifold topology.