Distances Between Products of Torus Knots

Definitions and mathematical statement

The supplied record asks for exact values of a rectangle transformation cost and a torus-knot cobordism distance, written respectively as \[ E(a\times b,c\times d) \qquad\text{and}\qquad d_{\chi}\left(T(a,b),T(c,d)\right). \] Here $a\times b$ and $c\times d$ denote the rectangles referred to in the supplied background, and $T(a,b)$, $T(c,d)$ denote the torus objects called knots in that record. The parameter domain and admissibility conditions are not supplied. Under the usual positive-integer convention, $T(p,q)$ is a torus knot when $\gcd(p,q)=1$, and otherwise is a torus link; this convention and any coprimality restrictions are not specified by the record and are not imposed here.

Requested determination. Determine the exact values of \[ E(a\times b,c\times d) \quad\text{and}\quad d_{\chi}\left(T(a,b),T(c,d)\right) \] for every admissible parameter quadruple $(a,b,c,d)$, once the intended domain and the definitions of both quantities have been recovered from an authoritative source.

Unresolved formulation. The record does not define the permitted rectangle transformations, their costs, or the conditions for a transformation to exist. It also does not define $d_{\chi}$: the ambient cobordism setting, regularity category, orientation conventions, allowed surface components, and the precise Euler-characteristic objective and normalization are unspecified. Thus the displayed request preserves the supplied target but does not determine a unique mathematical optimization problem.

Short English statement

Determine the exact rectangle transformation cost $E(a\times b,c\times d)$ and cobordism distance $d_{\chi}(T(a,b),T(c,d))$. The supplied record omits the parameter restrictions and definitions needed to make either quantity precise. Its open status is a supplied curation label, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30001539$, source problem code catalogue entry, title "Distances Between Products of Torus Knots," in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2010; no proposer is identified.
• Supplied report citation: Low-Dimensional Topology and Number Theory (2011), Oberwolfach Reports, pp. 2101--2163, DOI 10.4171/owr/2010/35. Original OWR report: https://doi.org/10.4171/owr/2010/35. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied citation has not been independently inspected here.
• Supplied literature source, labeled "stable four-genus results": https://arxiv.org/abs/0904.3054. No author or title is supplied for this reference.
• Literature assessment checked at 2026-08-21: this is supplied dated curation, not independently verified. It labels the problem open and states that the precise rectangle transformation cost and the precise cobordism distance between the indicated torus knots are not given by the cited report and remain nontrivial general optimization problems; later four-genus work gives only families and bounds. The supplied background calls this a dated literature triage derived from OWR curation and requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying verification says that the complete statement was preserved while mathematical notation was placed in delimiters. This records a supplied notation-preservation check and does not establish the missing definitions or mathematical correctness.