Kirby Problem 1.39

Definitions and mathematical statement

An algebraic knot is the oriented link $f^{-1}(0)\cap S^3_\epsilon$ of an irreducible complex plane curve singularity at $0\in\mathbb C^2$, for a sufficiently small sphere transverse to the curve. Use the orientation induced by the complex curve, and take distinct nontrivial oriented knot types as generators. Two knots are smoothly concordant if they cobound a smoothly embedded annulus in $S^3\times[0,1]$. Requiring instead a locally flat topological annulus defines topological concordance. Under connected sum the resulting groups are $\mathcal C$ and $\mathcal C^{\mathrm{TOP}}$; the inverse is the mirror with reversed orientation.

For each of these two groups, is the homomorphism \[ \bigoplus_{K\text{ algebraic}}\mathbb Z\longrightarrow\mathcal C \quad\text{or}\quad\mathcal C^{\mathrm{TOP}},\qquad e_K\longmapsto[K], \] injective? Explicitly, if distinct algebraic knots $K_1,\ldots,K_m$ and integers $a_1,\ldots,a_m$ satisfy $\sum a_i[K_i]=0$, must all $a_i$ be zero? The relation is finite, and negative coefficients mean taking concordance inverses. The smooth and topological assertions are separate questions, since the permitted annuli and slice disks differ.

Short English statement

Are the distinct algebraic knots linearly independent in smooth concordance, and also in topological concordance?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2698 (KP-1.39), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.39. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.