Kirby Problem 1.89

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.

A long $m$-component link is a smooth embedding $L:\bigsqcup_{j=1}^m\mathbb R\hookrightarrow\mathbb R^3$ agreeing with prescribed affine lines outside a fixed box. Let $\operatorname{Emb}_{\mathrm{long}}$ be the space of these parametrized embeddings with its smooth topology, and let $\operatorname{Emb}_{\mathrm{long}}(L)$ denote the path component containing $L$. For a closed link $L:\bigsqcup_{j=1}^m S^1\hookrightarrow M^3$, write $\operatorname{Emb}_M(L)$ for its component in the smooth embedding space. Components and parametrizations are fixed in these definitions; the spaces are not divided by ambient rotations.

A string link is a collection of properly embedded arcs in a cylinder with prescribed end points and collars. Infection data specify a trivial collection of strands in a neighborhood of disks meeting a link and a replacement string link with matching ends; replacing the trivial strands by the prescribed string link produces an infected link. Denote a specified such construction by $L=I(L_1,L_2)$, with the matching and gluing data included in $I$.

Statement recorded in the supplied source.

Determine the homotopy type of $\operatorname{Emb}_{\mathrm{long}}(L)$ for each long link $L$. Determine the homotopy type of $\operatorname{Emb}_M(L)$ for each link in a 3-manifold $M$. Given a compatible specified infection $L=I(L_1,L_2)$, describe the component of $L$ in the appropriate link embedding space in terms of the components of $L_1,L_2$ and the infection data. [S2]

Short English statement

Determine components of long-link and arbitrary 3-manifold link embedding spaces up to homotopy, and explain how these components behave under specified string-link infection.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2748 (KP-1.89). 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 1.89 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.