Symplectic Measure of Two-Component Link Types

Definitions and mathematical statement

Let $[L]$ denote a two-component link type, and let $\mathcal{R}([L])$ denote its collection of admissible geometric representatives $L=K_1\sqcup K_2$, with admissibility and equivalence understood as in the cited source. The supplied record calls $A(K_1,K_2)$ the symplectic measure of a representative but does not define this functional. Formally, the link-type quantity under consideration is \[ A([L]) := \inf\left\{A(K_1,K_2):K_1\sqcup K_2\in\mathcal{R}([L])\right\}. \] The requested problem is to determine $A([L])$ for two-component link types and, in particular, to decide whether \[ \forall\,[L]\text{ a hyperbolic two-component link type},\qquad A([L])>0. \] Here the infimum is over representatives of each fixed link type; the positivity question is posed separately for every hyperbolic type.

Unresolved qualifications. This is a formal transcription of the intended question, not a complete definition of its invariant. The supplied record does not specify the formula or normalization for $A(K_1,K_2)$, its domain and range, the ambient space, the regularity of representatives, or the equivalence relation defining link type. It also does not define the intended hyperbolicity condition. These source-dependent choices must be recovered before the displayed question can be treated as a fully specified mathematical statement.

Short English statement

For each two-component link type, take the infimum of the symplectic measure over all its admissible representatives. Determine this infimum and ask whether it is strictly positive for every hyperbolic two-component link type. The supplied record omits the definition of the measure and the precise admissibility and hyperbolicity conventions, so the statement requires source clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002333$, source problem code catalogue entry, titled Symplectic Measure of Two-Component Link Types, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2013 and names no proposer.
• Supplied source citation: Geometric Knot Theory (2014), Oberwolfach Reports, pp. 1313--1358, DOI 10.4171/owr/2013/22. The literature-source list also calls this Oberwolfach Report 22/2013. https://doi.org/10.4171/owr/2013/22. The report is described as documenting workshop discussions and open problems.
• Supplied literature reference: O'Hara, Measure of a 2-component link. https://doi.org/10.2748/tmj/1378991024.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem open and attributes to O'Hara a characterization of the zero minimum for a fixed geometric link and known zero-infimum classes, while describing positivity of $A([L])$ for hyperbolic links as open. This is supplied dated curation derived from the OWR record, not independently verified literature assessment. The cited claims and time-sensitive status require independent verification.