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Linking-form splitting and four-dimensional genera of knots

This paper refutes the general validity of Gilmer's splitting lemma and its applications to 4-dimensional knot genera by constructing counterexamples for specific primes, while subsequently establishing corrected versions of the lemma that hold under conditions such as Witt equivalence, cube-free orders, and for orientable surfaces of genus at least two.

Original authors: Seong-Jin Lee

Published 2026-10-30

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Seong-Jin Lee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there is a branch called topology that studies shapes not by their size or angles, but by how they can be stretched, twisted, and connected without tearing. Imagine a piece of clay that can be molded into any form, as long as you do not cut it or glue separate pieces together. Within this field, mathematicians are fascinated by knots: closed loops of string tangled in three-dimensional space. A central question is how these knots behave when they are pulled into a fourth dimension. Specifically, researchers ask whether a knot can be untangled smoothly in this higher space, or if it must pass through itself, creating a messy intersection. To answer this, they look for the simplest possible surface that the knot can bound in this four-dimensional world. If the surface is a simple disk, the knot is considered "slice" and perfectly unknotted in the higher dimension. If the surface is more complex, like a twisted band or a punctured torus, the knot is more complicated, and the "genus" of that surface—a measure of its complexity—tells us how tangled the knot truly is.

For decades, mathematicians have relied on a powerful tool called the linking form to estimate this complexity. Think of the linking form as a unique fingerprint left behind by a knot when it is viewed through a special mathematical lens. This fingerprint encodes deep information about the knot's structure. A famous rule, known as the splitting lemma, suggested that this fingerprint could always be broken down into two distinct parts: a simple, predictable part that corresponds to the surface the knot bounds, and a messy, chaotic part that can be ignored. This rule was the foundation for many important theorems, allowing mathematicians to prove that certain knots were too complex to be simple slices, or to calculate the minimum complexity of the surfaces they could bound. It was a trusted map for navigating the hidden geometry of knots.

However, a new study by Seong-Jin Lee has revealed that this trusted map is flawed. The researcher has constructed specific examples of knots that break the old rule, showing that the fingerprint cannot always be split in the way previously believed. The paper demonstrates that for certain knots, the mathematical structure is so intricate that it resists being divided into the simple and messy parts as the old theory predicted. This discovery does not just correct a minor detail; it overturns the conclusions of several major theorems that had been built upon the assumption that the splitting always worked. The author shows that the old rule fails precisely for a specific, rare class of knots, those whose mathematical fingerprints contain a particular kind of hidden symmetry that the old theory could not account for.

To prove this, the author did not rely on computer simulations or vague arguments, but constructed concrete examples using a type of knot known as a pretzel knot. These are knots formed by twisting strands of string in a specific, braided pattern. By carefully choosing the number of twists in each section of the knot, the researcher created two new types of knots. The first type bounds a smooth, non-orientable surface called a Möbius band, which is a one-sided surface like a strip of paper with a half-twist. The second type bounds a smooth punctured torus, a surface shaped like a donut with a hole in it. In both cases, the knots were designed to have a very specific, complex mathematical fingerprint. When the author analyzed these fingerprints, they found that the old splitting rule simply did not apply. The fingerprint could not be broken down into the expected simple part and the messy part. Instead, the structure remained whole and indivisible in a way that the previous theory claimed was impossible.

The implications of this finding are significant because it forces a re-evaluation of how mathematicians determine the complexity of knots. The old rule had been used to set lower bounds on how complex a surface must be to bound a given knot. Since the rule is now known to be false in certain cases, those lower bounds are no longer guaranteed to be correct for all knots. The author shows that for the specific knots constructed in the paper, the old method would have predicted a certain level of complexity, but the actual complexity is different. This means that previous calculations for these specific knots, and any knots with similar properties, may have been incorrect. The study identifies exactly which knots cause this problem: those whose mathematical fingerprints involve a specific type of prime number that leaves a remainder of three when divided by eight. For these knots, the old splitting rule fails completely.

Despite this failure, the paper does not leave mathematicians without a way forward. The author determines exactly what parts of the old theory remain true and what new rules can replace the broken ones. It turns out that the old rule still holds for most knots, specifically those whose mathematical fingerprints do not involve these rare, problematic prime numbers. For the knots where the rule fails, the author provides a new, more accurate way to measure complexity. This new method involves a different kind of mathematical invariant, a value that can be calculated directly from the knot's fingerprint without needing to split it. This new value provides a reliable lower bound for the complexity of the surface, even when the old splitting method fails. The author proves that this new bound is the best possible one that can be derived solely from the knot's fingerprint.

The study also clarifies the relationship between different types of surfaces. While the old theory suggested that the complexity of the surface was directly tied to the size of the simple part of the fingerprint, the new results show that this connection is more subtle. For knots that bound a Möbius band, the old theory would have predicted a certain structure, but the new examples show that the structure can be much more complex. Similarly, for knots that bound a punctured torus, the old theory failed to account for the full range of possibilities. The author's work shows that while the old splitting lemma was a useful guide, it was not a universal law. It worked for many cases, but it missed a crucial, hidden layer of complexity that only appears in specific, rare situations.

In the end, this paper serves as a necessary correction to the mathematical record. It takes a tool that was considered fundamental and shows its limits, replacing it with a more robust understanding. The author does not discard the old work but refines it, showing where it applies and where it does not. By constructing these specific counterexamples, the researcher has provided a clear boundary between what is true and what is not. The result is a more accurate picture of the four-dimensional world of knots, one where the complexity of the shapes is understood with greater precision. The study confirms that while mathematics often seeks simple, universal rules, the reality of these shapes is rich with exceptions that demand a more nuanced approach. The old map is no longer valid for the entire territory, but a new, more detailed map has been drawn for the areas where the old one failed.

Technical Summary: Linking-Form Splittings and Four-Dimensional Genera of Knots

Problem Statement
The paper addresses a foundational issue in low-dimensional topology regarding the relationship between the linking form of a knot's double branched cover and the topology of surfaces bounding that knot in the 4-ball. Specifically, it challenges the validity of "Gilmer's splitting lemma" and the subsequent rank criteria derived from it by Murakami–Yasuhara and Gilmer–Livingston. These results assert that if a knot $K$ bounds a surface $F$ in $B^4$ , the linking form $\lambda_K$ of the double branched cover $\Sigma_K$ admits an orthogonal splitting $\lambda_K \cong \lambda_Q \oplus \mu$ , where $\mu$ is metabolic and $Q$ is a form of rank and signature determined by the intersection form of the double cover of $B^4$ branched along $F$ . The paper investigates whether this splitting holds generally or if it fails under specific arithmetic conditions, and determines the extent to which the derived lower bounds for the nonorientable 4-genus $\gamma_4(K)$ and the topological 4-genus $\gamma_4^{top}(K)$ remain valid.

Methodology
The author employs a combination of algebraic topology, lattice theory, and knot theory:

Algebraic Analysis of Linking Forms: The paper utilizes the Witt group of linking forms on groups of odd order. It analyzes the Jordan decomposition of these forms and introduces the concept of "exceptional primes" (primes $p \equiv 3 \pmod 4$ where the odd levels of the $p$ -primary part of the linking form have specific rank and discriminant properties).
Construction of Counterexamples: To disprove the general validity of the splitting lemma, the author constructs specific families of knots using band moves on pretzel knots. These constructions involve:
- Pretzel knots $P(p^3, -p^3, p)$ and $P(p^3, -p^3, -p^2-p)$ for primes $p \equiv 3 \pmod 8$ .
- Band moves that transform these into knots bounding smooth M"obius bands and smooth punctured tori.
Kirby Calculus and Intersection Forms: The author computes the intersection forms of the double branched covers of these surfaces using Kirby diagrams and Goeritz matrices. This allows for the explicit calculation of the linking forms $\lambda_K$ and the verification of their non-splitting properties.
Lattice Existence Theorems: The paper uses the existence theorems for integral lattices with prescribed local invariants (determinant, signature, and $p$ -excess) to establish the existence of knots with specific linking form properties and to define new invariants.

Key Contributions and Results

Refutation of the Splitting Lemma: The paper proves that Gilmer's splitting lemma, as stated in [11, Lemma 1] and [24, Lemma 1.3], is false in general.
- Theorem A: For every prime $p \equiv 3 \pmod 8$ , there exist knots $K_p$ bounding smooth M"obius bands and knots $\hat{K}_p$ bounding smooth punctured tori (with clasp number one) whose linking forms do not admit an orthogonal summand on a group with at most two generators with a metabolic complement. This directly contradicts the rank criteria used to obstruct such surfaces.
- Counterexamples: The knots $K_p$ and $\hat{K}_p$ have determinants $p^9$ . Their linking forms possess "exceptional primes" where the splitting fails because the relevant subgroups are not direct summands of the homology group.
Conditions for Validity (Theorem B & C): The paper determines the precise conditions under which the splitting holds:
- Witt Equivalence: The lemma holds up to Witt equivalence for all knots.
- Cube-Free Determinants: The splitting holds as stated if the determinant of the knot is cube-free (i.e., no prime $p$ satisfies $p^3 \mid \det K$ ).
- Exceptional Primes: The failure of the splitting is entirely characterized by the presence of "exceptional primes." If no prime is exceptional for $\lambda_K$ , the splitting holds for nonorientable surfaces of genus $b_1(F) \ge 4$ and orientable surfaces of genus $g \ge 2$ .
New Invariants and Bounds:
- $h_\lambda(K)$ : The author defines $h_\lambda(K)$ as the least rank of an odd symmetric integral matrix of odd determinant whose discriminant form is Witt equivalent to $\lambda_K$ .
- Theorem D: It is proven that $h_\lambda(K) \le \gamma_4^{top}(K) \le \gamma_4(K)$ and that $h_\lambda(K) \le 3$ . Furthermore, $h_\lambda(K)$ is the optimal lower bound for $\gamma_4^{top}(K)$ that depends only on the Witt class of $\lambda_K$ .
- Comparison with $h_\oplus(K)$ : The paper defines $h_\oplus(K)$ as the least rank required for an orthogonal splitting (as opposed to Witt equivalence). It shows that $h_\lambda(K) \le h_\oplus(K) \le h_\lambda(K) + 2$ , with the gap occurring only at exceptional primes.
Optimal Bounds for Genus:
- Theorem F: Provides explicit local conditions (involving Legendre symbols, signature, and Arf invariant) to determine if $h_\lambda(K) \ge 2$ or $h_\lambda(K) \ge 3$ . This refines previous criteria for obstructing M"obius bands and punctured Klein bottles.
- Theorem G: Establishes that the rank criterion derived from the splitting lemma exceeds the true topological bound $h_\lambda(K)$ by at most 2, and only at exceptional primes.

Significance and Claims
The paper claims to correct a significant error in the literature regarding the application of linking forms to 4-dimensional genera.

Correction of Literature: It demonstrates that results relying on the strict orthogonal splitting of linking forms (such as Murakami–Yasuhara's bound on the clasp number and Gilmer's slice genus theorem for genus 1) are false in the presence of exceptional primes.
Scope of Validity: It clarifies that these results remain valid for knots with cube-free determinants (which includes all knots with up to 14 crossings, as noted in the text) and for higher genus surfaces where the rank constraints are less severe.
Optimality: The paper establishes $h_\lambda(K)$ as the sharpest possible lower bound for the topological nonorientable 4-genus that can be derived solely from the Witt class of the linking form.
Counterexamples: By constructing explicit knots bounding smooth M"obius bands and punctured tori that violate the rank criteria, the paper provides definitive counterexamples to the previously accepted theorems, necessitating a revision of how these invariants are applied in obstruction theory.

The author explicitly states that the proofs do not rely on computer calculations, though software was used to verify specific arithmetic examples. The work is supported by the Institute for Basic Science (IBS).

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