Conjecture 2.2 — A(S1; Z) is torsion free.

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 Jacobi diagram on an oriented circle consists of a finite uni-trivalent graph with univalent vertices attached to the circle and a cyclic ordering of the three incident half-edges at each trivalent vertex. Diagrams are considered up to orientation-preserving identification of the circle and graph. Its degree is half the total number of univalent and trivalent vertices. Let $\mathcal A(S^1;R)$ be the graded $R$-module they generate modulo AS, IHX and STU: reversing a trivalent cyclic order changes sign (AS); the three local reconnections of four external half-edges satisfy the oriented Jacobi identity (IHX); the difference between the two orders of neighboring legs on the circle equals the diagram joining them at a trivalent vertex (STU). All relations keep the rest of the diagram fixed. The framing-independence relation FI additionally sets to zero a diagram with an isolated chord whose two endpoints are consecutive on the circle. Write $\mathcal A_d$ for degree $d$.

Statement recorded in the supplied source.

Is $\mathcal A(S^1;\mathbb Z)$ torsion-free, equivalently is every degree piece torsion-free? Here no FI quotient is imposed: the question is for the integral AS--IHX--STU diagram module itself. Torsion-free means $m x=0$, $m\in\mathbb Z\setminus\{0\}$, implies $x=0$. [S2]

Short English statement

Is the integral circle Jacobi-diagram module without framing-independence quotient free of torsion?

Sources

• S1 ULAM. AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400024 (AMR-103-0024). 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 2.2 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.