This paper presents a significant advancement in the understanding of 2-knots by providing the first explicit examples of 2-knots that share the same knot group but possess distinct knot quandles. This addresses a long-standing open problem in knot theory, where such distinctions were known for 1-knots (e.g., the square knot and granny knot) but had no known analogues in higher dimensions.
The core innovation lies in leveraging and extending the concept of the “quandle type” as a powerful invariant. Knot quandles are algebraic structures that capture more detailed topological information about a knot than its knot group (the fundamental group of the knot’s complement). While an isomorphism between knot quandles implies an isomorphism between their associated knot groups, the converse is not generally true. This paper meticulously demonstrates this for 2-knots.
The key innovations and their integration are as follows:
Introduction of the Quandle “Type”: The authors introduce a specific algebraic invariant for any quandle, called its “type.” Defined as the minimum positive integer \(n\) (if it exists) such that for any two elements \(x, y\) in the quandle, iterating a specific binary operation \(n\) times (\(x *^n y\)) returns \(x\). If no such integer exists, the type is infinite. This “type” serves as a quantitative measure to distinguish quandles.
Calculation of Quandle Type for Generalized Alexander Quandles: A crucial technical result (Proposition 2.1) establishes that for a generalized Alexander quandle GAlex(\(G, f\)) (defined by a group \(G\) and its automorphism \(f\)), its type is precisely the order of the automorphism \(f\). This connects the abstract quandle invariant to a fundamental group-theoretic property.
Determining Monodromy Order for Twist Spins: The paper then focuses on a specific class of 2-knots called n-twist spins, denoted \(\tau^n(k)\), which are constructed from 1-knots \(k\) by Zeeman’s twist-spinning process. A critical step (Proposition 3.2, with proof in Appendix A) proves that the order of the monodromy automorphism \(\phi\) associated with the complement of \(\tau^n(k)\) is exactly the twist number \(n\).
Connecting Twist Number to Quandle Type: By combining the above, the paper proves a central theorem (Theorem 3.3): the type of the knot quandle of an n-twist spun knot \(\tau^n(k)\) is precisely \(n\). This directly links a topological construction parameter (\(n\)) to an algebraic invariant of the knot quandle, making the twist number a direct discriminator of quandles.
Construction of Counterexamples: Using the derived property, the authors construct infinitely many triples of 2-knots that fulfill the desired conditions. They consider twist spins of torus knots. It is a known prior result (from Gordon’s work) that for coprime integers \(p, q, r\), the knot groups of \(\tau^p(t_{q,r})\), \(\tau^q(t_{r,p})\), and \(\tau^r(t_{p,q})\) are mutually isomorphic (due to homeomorphic properties of their underlying branched covering spaces). However, by Theorem 3.3, their knot quandles have types \(p, q,\) and \(r\) respectively. Since \(p, q, r\) are coprime, their quandles must be non-isomorphic, thus providing the desired examples.
As a significant byproduct, the paper also provides a comprehensive classification of twist spins whose knot quandles are finite, both for oriented and unoriented 2-knots (Theorems 4.1 and 4.5). This classification unifies and extends previous partial results in the literature, demonstrating the utility of the quandle “type” as a robust invariant for distinguishing these complex 2-knot structures.
Main Prior Ingredients Needed: