Question 2.8 — (T.

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 finite group $G$, put $h_G(K)=\#\operatorname{Hom}(\pi_1(S^3\setminus K),G)$, counting homomorphisms themselves rather than conjugacy classes. A finite quandle is a finite set $X$ with a binary operation $\triangleright$ that is idempotent, has bijective right translations, and satisfies right self-distributivity. The knot quandle is the quandle of oriented meridians with its usual conjugation operation, or equivalently the quandle presented by arc labels and crossing relations. Put $h_X(K)=\#\operatorname{Hom}_{\mathrm{quandle}}(Q(K),X)$.

Statement recorded in the supplied source.

For finite groups $G$ other than dihedral groups, can $h_G$ be approximated pointwise by finite-type invariants? Seek one sequence $(v_j)$ of real-valued finite-type invariants, with degrees allowed to grow, for which $v_j(K)\to h_G(K)$ for every knot $K$. Investigate also the source's weaker query whether finite-type invariants convey any information about $h_G$ if full approximation fails, and its refinement replacing $h_G$ by $h_X$ for finite quandles $X$. [S2]

Short English statement

Can finite-type invariants approximate finite-group representation counts of knot groups, and the analogous finite-quandle coloring counts; if not, can they still provide information about those counts?

Sources

• S1 ULAM. AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400030 (AMR-103-0030). 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 2.8 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.