Strongly Negative Amphichiral Knots

• Strongly negative amphichiral knots are defined as smooth knots in S³ that admit an orientation-reversing involution fixing exactly two points on the knot.
• They exhibit a distinctive splitting of the Alexander–Conway polynomial and serve as key examples in the study of knot symmetry, rational concordance, and equivariant 4-manifold topology.
• These knots provide actionable insights into diagrammatic criteria, invariants such as Ozsváth–Szabó d-invariants, and generate nontrivial elements in smooth knot concordance groups.

A strongly negative amphichiral knot is an oriented knot $K \subset S^3$ admitting an orientation-reversing involution $\tau \colon S^3 \to S^3$, such that $\tau(K) = K$, $\tau^2 = \mathrm{id}$, and the fixed point set of $\tau$ consists of exactly two points on $K$. This symmetry property is distinguished from weak amphichirality by the existence of a strong involution fixing nontrivial points and reversing the orientation of both the ambient 3-sphere and the knot itself. Such knots form a principal family in knot symmetry classification, with deep connections to equivariant 4-manifold topology, rational concordance, and polynomial invariants.

1. Formal Definition and Symmetry Classification

Let $K:S^1 \to S^3$ be a smooth, oriented knot embedding. An involution $\tau$ of $(S^3, K)$ is a diffeomorphism $\tau:S^3 \to S^3$ of order 2 with $\tau(K) = K$. The involution is strongly negative amphichiral if (i) $\tau$ reverses the orientation of $S^3$, (ii) acts on the knot $K$ by orientation reversal ($\operatorname{deg}(\tau|_K) = -1$), and (iii) $\tau$ fixes exactly two points on $K$, so $\operatorname{Fix}(\tau;S^3) \cong S^0$ and $\operatorname{Fix}(\tau;K) \cong S^0$ (Boyle et al., 2023).

The geometrization theorem and orthogonal representation theory imply that symmetry groups of knots in $S^3$ are cyclic or dihedral, realized up to conjugacy via isometries. For the group $G \cong C_2$, there are six nontrivial symmetry types, one of which is strongly negative amphichiral (Table C2 in (Boyle et al., 2023)). In the dihedral case $G \cong D_n$, the “SNAP” (strongly negative amphichiral periodic) and “SNASI” (SNAP plus strongly invertible periodic) families represent prime knot symmetries exhibiting this property via rotation–reflection actions.

The canonical model for such an involution is the point-reflection $x \mapsto -x$ in $S^3 \subset \mathbb{R}^4$, fixing antipodal points and reversing all orientations.

2. Invariants, Criteria, and Factorization Properties

A central algebraic consequence of strong negative amphichirality is the splitting of the Alexander–Conway polynomial. For a strongly negative amphichiral knot, the Conway polynomial factors as $C_K(z) = \phi(z)\phi(-z)$ for some $\phi(z)\in \mathbb{Z}[z]$ (Conant et al., 2016, Boyle et al., 2022). Hartley–Kawauchi’s theorem proves this factorization for all such knots, and Conant–Manathunga’s work extends it modulo 4 for periodically amphichiral knots.

Further, Boyle–Chen introduce the half-Conway polynomial $\nabla_{(K,\rho)}(z)$ as the normalized factor $\phi(z)$, with precise skein relations for computation. Every integral polynomial $f(z)$ with $f(0) = 1$ arises as the half-Conway polynomial of some strongly negative amphichiral knot (Boyle et al., 2022).

Linking number, rotation number, and orthogonal-representation tests provide diagrammatic or topological criteria: for instance, the linking number around the involution axis must be odd for SNAc knots, and the representing involution in $O(4)$ must have exactly two −1 eigenvalues (Boyle et al., 2023).

3. Sliceness, Rational Concordance, and Canonical 4-Manifolds

Kawauchi proved that every strongly negative amphichiral knot bounds a smoothly embedded disk in a rational homology 4-ball $V_K$ (Prisa et al., 25 Sep 2025, Levine, 2022). Boyle–Chen established that, in the case of trivial Alexander polynomial, the disk can be made equivariant with respect to the involution: any SNAc knot with $\Delta_K(t)=1$ is equivariantly topologically slice in $B^4$ (Boyle et al., 2022). The proof utilizes surgery theory (Pin^− bordism, quadratic forms, Wall’s L-groups), equivariant handle calculus, and disk embedding techniques.

Levine and subsequent authors show that all such rational balls $V_K$ are diffeomorphic to a single manifold $Z_0 \setminus B^4$, and connected sums account for all known rationally slice but not slice knots (Levine, 2022). Thus, SNAc knots provide a universal source for nontrivial elements in the rational slice concordance kernel.

The rational sliceness can be detected structurally: for links preserved componentwise under the amphichiral involution, there are explicit JSJ decompositions and splicing conditions distinguishing when a knot is (or is concordant to) a strongly negative amphichiral knot (Prisa et al., 25 Sep 2025).

4. Obstructions: Determinant, Spin$^c$ Structures, and Floer Theory

Obstruction theory for SNAc knots bounding equivariant slice disks leverages classical and modern invariants. The determinant of a SNAc knot, $\det(K) = |H_1(\Sigma_2(K);\mathbb{Z})|$, must be a sum of two squares; for equivariantly slice knots, $\sqrt{\det(K)}$ must also be a sum of two squares (Boyle et al., 2021).

More refined obstructions arise from Donaldson’s theorem via the intersection forms on checkerboard surfaces, and compatibility with characteristic subgroups of Spin$^c$ structures. Failure of invariance under the lifted symmetry in the double branched cover prohibits equivariant sliceness, even if classical sliceness holds (Boyle et al., 2021).

Heegaard Floer correction terms (Ozsváth–Szabó $d$-invariants) provide further tests, requiring a unique fixed Spin$^c$ structure with $d=0$, and all other structures grouped into size-4 orbits with alternating signs (Boyle et al., 2021).

5. Canonical and Constructed Examples

The figure-eight knot $4_1$ is the paradigm for strong negative amphichirality, exhibiting all algebraic, diagrammatic, and sliceness properties. Symmetric diagrams under 180° rotation about a horizontal axis instantiate the required involution (Boyle et al., 2023, Conant et al., 2016).

Higher crossing knots are constructed via palindromic braids, infection and satellite operations on ribbon surfaces, and symmetrized surgery diagrams. For example, the knot $K(J)$ built by infecting the figure-eight Seifert surface bands with a reversible knot $J$ and its mirror is strongly negative amphichiral and algebraically slice (Miller, 2020). Infinite families are produced by iterated doubling, satellite, and splicing methods, with many explicit computations and counterexamples revealing the boundaries of sliceness and symmetry (Boyle et al., 2022, Boyle et al., 2022, Miller, 2020).

Among two-bridge knots and torus knots, specific arithmetic conditions (e.g., $q^2 \equiv -1 \pmod{p}$ for two-bridge $K(p/q)$; odd parameters for torus knots $T_{2,2m+1}$) produce SNAc examples (Boyle et al., 2023).

6. Concordance, Genus, and Algebraic Structure

Strongly negative amphichiral knots generate 2-torsion in the smooth knot concordance group: $K \# K$ is smoothly slice, but $K$ itself may not be (locally flat) slice in $B^4$ (Miller, 2020). Their topological 4-genus can be made arbitrarily large, disproving conjectures about the relationship between clasp number and genus (Miller, 2020). Large families have been constructed where, despite rational sliceness, the minimal genus of any locally flat surface bounded by $K$ in $B^4$ is arbitrarily high.

In the setting of equivariant $\mathbb{Q}$-concordance, these knots naturally generate nontrivial kernels—both in the abelianization and with respect to Milnor’s invariants. The development of an equivariant algebraic $\mathbb{Q}$-concordance group remains an active topic (Prisa et al., 2024).

7. Open Problems and Future Directions

Open questions pertain to the uniqueness of the rational homology balls for all rationally slice knots, existence of knots slice in $\mathbb{Z}_2$-homology balls not constructed from SNAc templates, smooth versus topological equivariant sliceness distinctions, existence of equivariant genus bounds, and full classification of possible half-Conway polynomials. The equivariant Floer-theoretic invariants and the structure of the equivariant concordance group (e.g., non-abelianity, solvability) are major frontiers (Prisa et al., 2024, Levine, 2022, Boyle et al., 2022).

Strongly negative amphichiral knots remain central in modern knot theory for their unique algebraic, topological, and equivariant properties, bridging deep areas of 3-manifold symmetries, 4-manifold topology, and algebraic concordance.