Problem 3.11 — (M.

Definitions and mathematical statement

Configuration-space integral constructions use compactified configuration spaces with boundary strata describing collisions of points. In a standard terminology, hidden collision strata involve larger clusters rather than only the principal pairwise faces; the precise strata in the three approaches referenced by the source are not supplied.

Find another way to eliminate the hidden-stratum contributions so that all three source approaches naturally become the mapping degree of one and the same map. For context, a degree interpretation for a map $F:X\to Y$ between closed oriented spaces of equal dimension satisfies $F_*[X]=\deg(F)[Y]$; configuration-space versions may require gluing, relative classes, or other specified conventions. The actual spaces, target, map, and boundary cancellation data are absent. The aim is a common geometric degree construction, not merely numerical equality of three unnamed invariants.

The extract records two qualified pieces of background: S. Poirier [328] related the first and second approaches using quasi-tangles under a vanishing-anomaly assumption, with a cross-reference to Question 3.12; D. Thurston suggested Etingof--Kazhdan $R$-matrices [117] as potentially useful for comparing the first and third. The former is conditional on the stated anomaly hypothesis and the latter is a suggestion, not an asserted equivalence. Neither is promoted into the requested alternative hidden-stratum construction. The following Section 3.6 heading is separate material.

Short English statement

Eliminate hidden configuration-space boundary contributions in a new way that unifies the source's three constructions as the degree of the same map. Their definitions and the exact boundary/map data are missing; the stated Poirier comparison remains conditional on vanishing anomaly.

Sources

• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 3.11, PDF page 56, attributed to M. Polyak, https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf; accessed 2026-07-29, PDF-font-structure extraction.
• S. Poirier [328], conditional first/second-approach comparison via quasi-tangles, and source Question 3.12; D. Thurston's suggestion involving Etingof--Kazhdan $R$-matrices [117]. The record supplies no bibliography entries for [328] or [117].
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400061$, source code catalogue entry, Problem 3.11. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.