Conjecture 2.1 — ([220, Problem 1.92 (N)]) Fd(ZK)/Fd+1(ZK) is torsion free for each d.

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 $F_{d+1}(R\mathcal K)$.

Statement recorded in the supplied source.

For every integer $d\ge0$, is the abelian group \[F_d(\mathbb Z\mathcal K)/F_{d+1}(\mathbb Z\mathcal K)\] torsion-free? This means that $m x=0$ with integer $m\ne0$ forces $x=0$. The statement records the source's conjecture; no present-day status claim is inferred from its historical formulation. [S2]

Short English statement

Does each graded piece of the integral finite-type knot filtration have no nonzero torsion elements?

Sources

• S1 ULAM. AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400023 (AMR-103-0023). 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 2.1 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.