Kirby Problem 1.56

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a smooth knot $K\subset S^3$, $\widehat{\mathrm{HFK}}(K)$ denotes the bigraded abelian group of hat knot Floer homology with integer coefficients, with its Alexander and Maslov gradings, as in the source’s basic convention. The isomorphism in the question respects these gradings. A ribbon concordance is a smooth embedded annulus in $S^3\times[0,1]$ whose height function can be chosen with no local maxima. Two knots are isotopic if an ambient smooth isotopy of $S^3$ carries one to the other.

Statement recorded in the supplied source.

If smooth knots $K_0,K_1\subset S^3$ are joined by a ribbon concordance and \[\widehat{\mathrm{HFK}}(K_0)\cong\widehat{\mathrm{HFK}}(K_1),\] must $K_0$ and $K_1$ be isotopic? [S2]

Short English statement

Does ribbon concordance together with equal graded knot Floer homology force the two knots to be isotopic?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2715 (KP-1.56). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, eds., K3: A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295, American Mathematical Society (2026), Problem 1.56 and its remarks; Chapters 1 and 2 for the stated conventions. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.