Kirby Problem 1.2

Definitions and mathematical statement

Let $V=S^1\times D^2$ be a solid torus and let a pattern $P\subset\operatorname{int}V$ be a smooth knot. For a knot $K\subset S^3$, identify $V$ with a tubular neighborhood of $K$, sending the preferred longitude of $V$ to the zero-linking longitude of $K$. The image of $P$ is the satellite $K_P$. Write $c(J)$ for the minimum number of crossings among diagrams of a knot $J$. The geometric winding number is \[ w(P)=\min_D |P\cap D|, \] where $D$ ranges over properly embedded meridian disks of $V$ transverse to $P$. These are unsigned intersection counts, so $w(P)$ can differ from the absolute value of the integer represented by $[P]\in H_1(V;\mathbb Z)$.

The two proposed inequalities are:

1. For a genuine satellite with nontrivial companion $K$ and essential companion torus, $c(K_P)\geq c(K)$.
2. More generally, for every pattern and companion, $c(K_P)\geq w(P)^2c(K)$.

Here an essential companion torus is incompressible and not boundary parallel in the satellite exterior. The second inequality uses the geometric winding number even for patterns with algebraic winding number zero.

Short English statement

Can a satellite have fewer crossings than its companion, or fewer than the companion's crossing number multiplied by the square of the pattern's geometric winding?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2661 (KP-1.2), 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 1.2. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.