Problem 2.9 — (X.-S.

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 a Seifert matrix $S$ of a knot, its classical signature $\sigma(K)$ is the signature of the symmetric matrix $S+S^{\mathsf T}$, for a fixed oriented Seifert-form convention; reversing this overall sign does not change this approximation question. The signature of a real symmetric matrix is its number of positive eigenvalues minus its number of negative eigenvalues.

Statement recorded in the supplied source.

Is there a single sequence $(v_j)_{j\ge1}$ of real-valued finite-type knot invariants, with no fixed degree bound, such that \[\lim_{j\to\infty}v_j(K)=\sigma(K)\] for every knot $K$? The limit is pointwise over knot types, not a claim that signature itself has finite type. [S2]

Short English statement

Is the knot signature a pointwise limit of a sequence of finite-type invariants?

Sources

• S1 ULAM. AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400031 (AMR-103-0031). 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 2.9 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.