Smooth and topological knot concordance — Examples and Counterexamples

Given: ${\mathrm{A}\mathrm{C}}_{\omega }$ and a smooth oriented knot $K\subset S^3$.

The reverse mirror is formed by applying the fixed orientation-reversing reflection and then reversing the knot orientation. (Reverse mirror of an oriented knot)

A smooth proper disk is locally flat as a topological proper surface, and its genus is $0$ because $\chi (D^2)=1$. (Locally flat proper surfaces and concordance collars, Smooth and topological slice genus)

The smooth and topological slice genera are the least genera in the respective nonempty surface classes. (Smooth and topological slice genus, Nonemptiness and minima for the smooth and locally flat slice genera)

The general reverse-mirror disk construction applies to every smooth oriented knot. (A knot summed with its reverse mirror bounds a ribbon disk)

In the smooth concordance group, the reverse-mirror class is the additive inverse; the current theorem's smooth branch proves this using the disk in [F4]. (Smooth and topological knot concordance classes form abelian groups)

The full TOP group interface likewise gives the reverse-mirror class as the inverse for locally flat concordance, with the same countable-choice hypothesis. Its TOP connected-sum input includes the compact real-code/$L[r]$ specialization in the compatibility supplier at exactly this standard-product annulus scope; the unqualified literature assertion alone is not treated as an AC_omega theorem. No new local normal-neighborhood proof is asserted here. (Smooth and topological knot concordance classes form abelian groups)

Given: The standard unknot $U$, the standard right-handed trefoil $T$, and the claim in the Statement refuted.

An abstract oriented cobordism is a compact oriented surface with boundary $-C_0\bigsqcup C_1$ and contains no ambient embedding data. (Abstract oriented cobordism of knot circles)

The Seifert matrix convention in the library is $V_{ij}=\mathrm{lk}(a_i^{+},a_j)$ for a chosen oriented surface and basis; the surface and basis are part of the data until invariance is proved. (Seifert pairing and matrix of an oriented knot)

A trefoil Seifert surface admits a rank-two Seifert matrix whose symmetric sum is $\left(\begin{array}{cc}-2& 1\\ 1& -2\end{array}\right)$; if the cited convention uses the transpose matrix, the symmetric sum is unchanged. For a real symmetric matrix, its signature is the number of positive eigenvalues minus the number of negative eigenvalues. (Collins, Theorem 2.2.9 and Example 2.2.13, https://era.ed.ac.uk/bitstream/handle/1842/5034/Collins2011.pdf?sequence=4)

The standard unknot bounds a disk in $S^3$; the disk is contractible, so its first homology is zero and its symmetrized Seifert form has signature $0$. (Oriented connected sum and prime knot)

A metabolizer for a rank-two integral Seifert form is a rank-one direct summand $L$ with $V(x,y)=0$ for $x,y\in L$. Thus every nonzero $x\in L$ satisfies $x^T(V+V^T)x=2V(x,x)=0$. This is immediate from the defining bilinear relation, with no signature-invariance or general signature-vanishing theorem. (Metabolic Seifert forms and algebraic knot concordance)

Under the Axiom of Countable Choice, a knot is smoothly slice if and only if it is smoothly concordant to the standard unknot. (Smooth sliceness is equivalent to concordance to the unknot)

Full Axiom of Choice is assumed by the supplied Seifert-pairing definition. (The Axiom of Choice)

Full Axiom of Choice implies the countable-choice hypothesis used by [F6]. (AC implies DC implies countable choice)

Every locally flat slice disk, including every smooth slice disk, induces a metabolizer for the positive-push Seifert form of each specified smooth Seifert surface. The proof uses a locally flat hypersurface and smooth chains only in disjoint smooth ambient neighborhoods; it requires no smoothing of that hypersurface. A slice disk induces a Seifert metabolizer, A locally flat slice surface bounds a Seifert hypersurface