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Casson–Gordon invariants and sliceness obstructions — Examples and Counterexamples

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

None yet.

4 · Examples, counterexamples and false statements

CounterexampleConstruction: Literature-sourcedVerification: AI-generatedverified 2026-10-22 (gpt-6.1-sol)Open item page →

A prime-power torsion linking form need not have a metabolizer

Statement refuted

Assume AC. The assertion that every nonsingular linking form on a finite group of odd prime-power order has a metabolizer is false: the double branched cover of the positive trefoil has cyclic first homology of order three and no metabolizer.

Statement references

The metabolizer definition

The oriented linking-pairing convention

The positive trefoil Seifert matrix

The branched-cover surgery presentation

The surgery linking form formula

Axiom of Choice

Rolfsen, Knots and Links, Chapter 8.D, pp. 212–215

Przytycki and Yasuhara, Linking Numbers in Rational Homology 3-Spheres, Cyclic Branched Covers and Infinite Cyclic Covers, §§1 and 3

Facts & Assumptions

Given: The positive trefoil, its Seifert matrix $V$ , and the double branched-cover orientation convention.

[A1]

Full AC is assumed wherever the cover and orientation conventions use the choice axiom. The Axiom of Choice

[F1]

$V = (\begin{array}{cc} - 1 & 0 \\ - 1 & - 1 \end{array})$ . A positive Hopf plumbing gives the trefoil Seifert matrix

[F2]

The double-cover surgery matrix is $A = V + V^{T}$ . Surgery presentation from a Seifert surface

[F3]

The oriented linking form on the surgery cokernel is $- A^{- 1} m o d Z$ . Inverse-matrix formula for a surgery linking form

[F4]

A finite nonsingular linking form has a metabolizer only if some subgroup has square-root order. Finite nonsingular linking forms and metabolizers

[F5]

The surgery linking matrix computes the same oriented torsion linking pairing as the signed bounding-chain formula. Bounding-chain expression for a torsion linking pairing Inverse-matrix formula for a surgery linking form

[F6]

A closed oriented $3$ -manifold with finite first homology has nonsingular torsion linking form. The torsion linking pairing of a rational homology sphere is nonsingular

Counterexample

1.1F1F2F3F5givenalgebraA1

Computing gives $A = (\begin{array}{cc} - 2 & - 1 \\ - 1 & - 2 \end{array})$ with determinant $3$ , and unimodular row and column operations reduce it to $diag (1, 3)$ . Therefore $H_{1} (Σ_{2} (T); Z) ≅ Z / 3$ . The matrix formula [F3] gives generator self-linking $2 / 3 \equiv - 1 / 3 m o d Z$ under [F5], so the full group is not isotropic.

2.1F4F6step 1.1algebra∎

The only subgroups of $Z / 3$ have order $1$ or $3$ , while a metabolizer would have order $\sqrt{3}$ by [F4]. The form is nonsingular by [F6] and step 1.1, so it satisfies the assertion's nonsingularity hypothesis. No subgroup has that order, so no metabolizer exists. This finite nonsingular form of odd prime-power order refutes the assertion.

ExampleConstruction: Literature-sourcedVerification: AI-generatedverified 2026-10-22 (gpt-6.1-sol)Open item page →

The square knot has a nontrivial character vanishing on a linking metabolizer

Example

Assume AC. For the square knot $K = T # T^{†}$ , where $T$ is the positive trefoil and $T^{†}$ is its reverse mirror, the double branched cover has $H_{1} ≅ Z / 3 \oplus Z / 3$ with linking form $diag (1 / 3, - 1 / 3)$ . Its diagonal subgroup is a metabolizer, and the order-three character $χ (a, b) = (a - b) / 3 m o d Z$ vanishes on it. The knot is smoothly slice.

Example references

A knot plus its reverse mirror is ribbon

The branched-cover surgery presentation

The surgery-matrix linking-form formula

Charles Livingston and Swatee Naik, Introduction to Knot Concordance, §3.1, pp. 44–51

Axiom of Choice

Rolfsen, Knots and Links, Chapter 8.D, pp. 212–215

The oriented bounding-chain linking convention

Finite-order characters

The oriented connected-sum convention

The smooth sliceness convention

Facts & Assumptions

Given: The positive trefoil $T$ , its reverse mirror $T^{†}$ , their boundary-connected-sum Seifert surface, the positive-push convention, and the double branched cover.

[A1]

Full AC is assumed wherever the cover and orientation conventions use the choice axiom. The Axiom of Choice

[F1]

A positive trefoil Seifert matrix is $V = (\begin{array}{cc} - 1 & 0 \\ - 1 & - 1 \end{array})$ . A positive Hopf plumbing gives the trefoil Seifert matrix

[F2]

The reverse-mirror surface is $- r (F)$ ; its positive normal is the reflected positive normal because the two orientation reversals cancel. Reflection negates linking, so the transported basis has positive-push matrix $- V$ . Seifert pairing and matrix of an oriented knot Well-definedness of the Seifert pairing Reverse mirror of an oriented knot

[F3]

For $q = 2$ , a Seifert matrix $U$ gives a surgery presentation of the double branched cover with symmetric matrix $U + U^{T}$ . Surgery presentation from a Seifert surface

[F4]

For a nonsingular integral surgery matrix $A$ , the first homology is $Z^{n} / A Z^{n}$ and the oriented linking form is $- A^{- 1} m o d Z$ . Inverse-matrix formula for a surgery linking form

[F5]

Every knot plus its reverse mirror bounds a smooth ribbon disk. A knot summed with its reverse mirror bounds a ribbon disk

[F6]

In a finite nonsingular linking form, an isotropic subgroup of square-root order is a metabolizer. Finite nonsingular linking forms and metabolizers

[F7]

For integral $1$ -cycles $a, b$ representing the two torsion classes and an integral $2$ -chain $C$ with $\partial C = n a$ , $n > 0$ , the oriented torsion linking pairing is $(C \cdot b) / n m o d Z$ , with $C$ transverse to $b$ . Bounding-chain expression for a torsion linking pairing

[F8]

A finite-order character is a homomorphism to $Q / Z$ , with order measured in the character group. Finite-order characters of a cyclic branched cover

[F9]

The oriented connected sum joins the oriented knot arcs in the fixed two-ball model; the boundary-connected-sum surface in the Given therefore represents $T # T^{†}$ . Oriented connected sum and prime knot

[F10]

A knot bounding a smooth proper disk in $B^{4}$ is smoothly slice. Smooth and topological slice genus

Verification

1.1F1F2F9givenconstructA1

The boundary-connected-sum surface for $K$ has block-diagonal Seifert matrix $U = V \oplus (- V)$ : curves in the two summands are disjoint, so mixed linking entries vanish. Hence $U + U^{T} = A \oplus (- A)$ , where $A = V + V^{T} = (\begin{array}{cc} - 2 & - 1 \\ - 1 & - 2 \end{array}) .$

2.1F3step 1.1constructalgebra

The double-cover surgery presentation [F3] gives the linking group as $coker (A) \oplus coker (- A)$ . Unimodular row and column operations reduce $A$ to $diag (1, 3)$ , so each cokernel is $Z / 3$ .

2.2F4F7step 1.1algebra

Since $A^{- 1} = - \frac{1}{3} (\begin{array}{cc} 2 & - 1 \\ - 1 & 2 \end{array})$ , [F4] gives self-linking $2 / 3 \equiv - 1 / 3$ on a generator of the original-trefoil block. The reverse-mirror block has matrix $- A$ , and therefore generator self-linking $- 2 / 3 \equiv 1 / 3$ . Choose the coordinates $(a, b)$ with the reverse-mirror block first and the original-trefoil block second. The mixed terms vanish, so in these explicitly ordered coordinates the linking form is $diag (1 / 3, - 1 / 3)$ , as asserted.

3.1F6F8givenstep 2.2algebra

The diagonal subgroup $P = {(a, a) : a \in Z / 3}$ has order $3$ and $λ ((a, a), (b, b)) = a b / 3 - a b / 3 = 0$ . The ambient group has order $9$ , so $∣ P ∣^{2} = 9$ and [F6] makes $P$ a metabolizer. By [F8], the map $χ (a, b) = (a - b) / 3 m o d Z$ is a character of order $3$ and vanishes on $P$ .

4.1F5F10given∎

By [F5], $T # T^{†}$ bounds a smooth ribbon disk and is smoothly slice by [F10]. This establishes sliceness independently of the homology and character calculation.

Diagram

The following is the positive trefoil as the closure of the two-strand braid $σ_{1}^{3}$ . Here $σ_{1}$ means the strand entering from the left passes under the strand entering from the right; the closure arcs connect endpoints in equal horizontal positions. Three crossings exchange the two endpoints, so the closure has one component. The proof uses the Seifert-matrix calculations above and does not rely on the drawing.

\begin{tikzpicture}[x=0.9cm,y=0.9cm,baseline=(current bounding box.center)]
  \draw[thick] (-1,3)--(1,2)--(-1,1)--(1,0);
  \draw[thick] (1,3)--(-1,2)--(1,1)--(-1,0);
  \fill[white] (0,2.5) circle (2.5pt);
  \draw[thick] (-0.28,2.36)--(0.28,2.64);
  \fill[white] (0,1.5) circle (2.5pt);
  \draw[thick] (-0.28,1.36)--(0.28,1.64);
  \fill[white] (0,0.5) circle (2.5pt);
  \draw[thick] (-0.28,0.36)--(0.28,0.64);
  \draw[thick] (-1,0)..controls(-1.7,0)and(-1.7,3)..(-1,3);
  \draw[thick] (1,0)..controls(1.7,0)and(1.7,3)..(1,3);
\end{tikzpicture}

The reverse mirror factor is obtained by reflecting the diagram and reversing its orientation; the square knot is their connected sum. The diagram only records the braid convention used to name $T$ .

Casson–Gordon invariants and sliceness obstructions — Examples and Counterexamples

2 · Logical flowchart

3 · Definitions, theorems and proofs

4 · Examples, counterexamples and false statements

A prime-power torsion linking form need not have a metabolizer

Statement refuted

Statement references

Facts & Assumptions

Counterexample

The square knot has a nontrivial character vanishing on a linking metabolizer

Example

Example references

Facts & Assumptions

Verification

Diagram

Sources