Kirby Problem 1.1

Definitions and mathematical statement

A knot is a smooth embedding of a circle in the oriented sphere $S^3$, considered up to ambient isotopy. A diagram is a generic projection to a plane, with an overpassing strand specified at every transverse double point. If $D$ is such a diagram, let $c(D)$ be its number of double points. The crossing number is \[ c(K)=\min\{c(D):D\text{ is a diagram of }K\}. \] For oriented knots $K_1,K_2$, their connected sum $K_1\#K_2$ is obtained by deleting a short unknotted arc from each and joining the resulting endpoints without introducing additional knotting. Its oriented isotopy class is independent of these choices.

The question is whether every pair of knots satisfies \[ c(K_1\#K_2)=c(K_1)+c(K_2). \] Joining minimal diagrams proves the upper bound. The requested content is the reverse inequality for an arbitrary diagram of the composite knot; that diagram need not visibly display its summands. The assertion includes nonalternating knots and arbitrary numbers of summands by induction.

Short English statement

Does tying two knots together always add their minimum crossing numbers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2660 (KP-1.1), 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 1.1. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.