Problem 12.6 — Find a new proof of the existence of a universal Vassiliev in- variant of knots, presenting them by KTG’s and their o…

Definitions and mathematical statement

A knotted trivalent graph (KTG), in the usual setting, is a trivalent graph embedded in $S^3$ with the ribbon/framing structure required by its operations; knot components may occur in the relevant extended category. The excerpt does not specify the elementary KTGs, operation list or presentation relations.

A Vassiliev invariant of order at most $d$ extends by crossing differences and vanishes on knots with more than $d$ double points. A universal invariant has a degree-completed diagram-valued expansion $Z(K)$ whose degree-$d$ truncation recovers every order-at-most-$d$ invariant by linear evaluation, with the intended symbol correspondence.

Problem 12.6 asks for a new proof of existence of such a universal invariant, using presentations of knots by KTGs and their operations. The proof must explain how the KTG presentations and their relations yield a well-defined knot invariant and establish universality, rather than simply invoke an existing construction. This is a requested proof method, not a claim that existence of universal Vassiliev invariants is itself an unsettled bare existence problem.

The source does not prescribe the coefficient field, framing/orientation category, diagram target, precise KTG operations or any correction/coherence conventions. Strict compatibility with an arbitrarily selected unmodified operation list is not added. The task retains its KTG route without inventing generators, relations or an operation-level invariant that has not been supplied.

Short English statement

Give a new KTG-based proof of existence of a universal finite-type knot invariant. The KTG generators, operations, relations and normalization must be specified; the goal is a new proof through this presentation method.

Sources

• Local UnsolvedMath registry, permanent ID 10400212, source code catalogue entry; no proposer or proposal year supplied; updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 12.6, PDF page 162: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Registry access date 2026-07-29; PDF font-structure extraction. Provenance distinguishes 231 numbered bold headings. Present primary-literature status and redistribution terms require review.
• The supplied summary repeats the KTG-based new-proof request without exact further references or a verified resolution.