Conjecture 2.5 — Vassiliev invariants distinguish oriented knots.

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A knot or link is a smooth oriented embedding of one or more circles in the oriented $3$-sphere, considered up to orientation-preserving ambient isotopy. Write $O$ for the unknot, which bounds an embedded disk. Let $R\mathcal K$ be the free $R$-module on oriented knot isotopy classes. Extend a knot invariant to singular knots with ordinary transverse double points by $v(K_\times)=v(K_+)-v(K_-)$. Let $F_d(R\mathcal K)$ be the submodule generated by the alternating resolution sums of knots with $d$ such points. An $R$-valued finite-type (Vassiliev) invariant of degree at most $d$ is an $R$-linear map $R\mathcal K\to R$ vanishing on $F_{d+1}(R\mathcal K)$. For coefficients in an abelian group $A$, use homomorphisms $\mathbb Z\mathcal K\to A$ that vanish on $F_{d+1}(\mathbb Z\mathcal K)$; this is the same finite-type definition with those coefficients.

Statement recorded in the supplied source.

Do finite-type invariants distinguish oriented knots? Precisely, for any two nonisotopic oriented knots $K,K'$, is there a finite degree and a finite-type knot invariant $v$ with $v(K)\ne v(K')$? The degree can depend on the pair. The query concerns oriented isotopy classes, not their unoriented classes. [S2]

Short English statement

Can some finite-type invariant distinguish every pair of distinct oriented knot types, allowing its degree to depend on the pair?

Sources

• S1 ULAM. AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400027 (AMR-103-0027). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 T. Ohtsuki (editor), Problems on invariants of knots and 3-manifolds, Geometry & Topology Monographs 4 (2002), pp. 377--572 (published 2004), item 2.5 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.