Problem 12.14 — (N.

Definitions and mathematical statement

For a knot $K\subset S^3=\partial B^4$, let $g_s(K)$ be its four-genus: in the smooth convention, the minimum genus of a smooth, connected, compact, oriented surface properly embedded in $B^4$ with boundary $K$. A slice knot bounds a properly embedded disk in the same category, equivalently has four-genus zero. A crossing switch interchanges the overpassing and underpassing arcs at one crossing in a knot diagram; ambient isotopies may be used between switches.

Askitas asks whether every knot $K$ can be changed to a slice knot using $g_s(K)$ crossing switches. Writing \[u_s(K)=\min\{n\geq0:\text{a sequence of }n\text{ crossing switches changes }K\text{ to a slice knot}\},\] the usual "using at most" interpretation is \[\forall K,\quad u_s(K)\leq g_s(K).\] The customary crossing-change cobordism inequality gives $g_s(K)\leq u_s(K)$, so in that convention the question is whether $u_s(K)=g_s(K)$ for all knots. Equivalently, seek a sequence $K=K_0,\ldots,K_n$ with $n\leq g_s(K)$, each step a crossing switch up to isotopy, and $g_s(K_n)=0$. The terminal knot may be any slice knot.

The source does not explicitly state the category for four-genus or slicing. A locally flat topological interpretation would replace the surfaces and disks throughout, and cannot be silently identified with the smooth question. It also does not clarify a literal "exactly" versus the customary minimal-number interpretation; the displayed formulation makes that choice visible rather than imposing diagram restrictions absent from the source.

Short English statement

Can every knot be made slice with as many crossing switches as its four-genus? In the usual minimal-number formulation this asks whether the slicing number always equals the four-genus, with a consistent smooth or topological category.

Sources

• Local UnsolvedMath registry, permanent ID $10400219$, exact source code catalogue entry.
• N. Askitas, attribution in Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 12.14, PDF page 166: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Registry access 2026-07-29, pdf-font-structure extraction.