Deep derived pure braids need not be 1-solvable

Abstract

For every m≥3 and every finite k≥1, the kth ordinary derived subgroup of the pure braid group contains a braid whose standard closure is not integrally smoothly 1-solvable. Consequently P_m^{(n+2)}⊄F_n^m for every integer n≥1, answering negatively the derived-series containment question in the Harvey–Park–Ray Oberwolfach contribution. The proof combines established signature-quasimorphism results with amenable-quotient rigidity and a pointwise signature obstruction to 1-solvability. The general quasimorphism ingredients are prior results. An accompanying exact verification package also preserves an explicit 104-crossing example in P_3^{(3)}.

The idea in plain English

Pure braids return each strand to its starting position. Repeatedly taking commutators creates the derived series, an increasingly restrictive sequence of subgroups. A separate notion, called 1-solvability, asks whether the closed braid fits into a four-dimensional construction with specified surfaces and intersection properties.

The paper shows that for every number of strands at least three, a braid can lie arbitrarily far down the derived series and still fail 1-solvability. For each finite depth there is a witness; the argument does not use one braid that works simultaneously at every depth.

The general argument connects established results about braid signatures and quasimorphisms. The supporting materials also give a concrete three-strand word with 104 crossings at the third derived level. That finite example illustrates one case of the broader existence theorem.

The precise result

Let P_m be the classical disk pure braid group, with P_m^(0)=P_m and P_m^(k+1)=[P_m^(k),P_m^(k)]. For every m≥3 and every finite integer k≥1, some braid in P_m^(k) has standard oriented closure that is not integrally smoothly 1-solvable in the sense of Otto Definition 2.3.

Equivalently, the string-link concordance classes represented by P_m^(k) are not all in F₁^m. Since F_n^m⊂F₁^m for integers n≥1, this gives P_m^(n+2)⊄F_n^m for every m≥3 and integer n≥1, a negative answer to Question 3 in the Harvey–Park–Ray Oberwolfach contribution.

For the explicit third-derived example, set a=σ₁², b=σ₂², c=[a,b], d=a⁻¹ca, e=b⁻¹cb, u=[c,d], v=[d,e] and β=[u,v], using [x,y]=xyx⁻¹y⁻¹. The paper gives a 104-letter freely reduced Artin word for β with closure signature −2.

Why the result holds

The ordinary signature s of a three-strand braid closure has bounded failure of additivity. Its homogenization q is a homogeneous quasimorphism, exactly linear on powers. Published signature formulas show that q restricted to the pure braid group P₃ is not a homomorphism.

A nonhomomorphic homogeneous quasimorphism cannot vanish on a finite derived subgroup. Otherwise it would descend to a solvable, hence amenable, quotient, where homogeneous quasimorphisms are homomorphisms. The general ingredients are established results.

The ordinary signature and its homogenization are distinct. A suitable positive power is essential: nonzero q on a braid does not by itself give nonzero ordinary signature on that same braid. The powered braid has nonzero ordinary signature and zero pairwise linking numbers. A pointwise signature obstruction rules out an integral smooth 1-solution; adding split trivial strands reaches every m≥3.

Scope and limitations

The result uses the ordinary derived series of classical disk pure braid groups and integral smooth solvability of standard closures. It makes no claim for power subgroups, lower central series, rational derived series, surface braid groups, rational solvability or a merely topological solvability category.

The witness may depend on the finite depth k. No single braid in the intersection of all derived subgroups is asserted, and the explicit 104-crossing word is supplied only for k=3. No minimal crossing number or effective word-length bound at arbitrary depth is claimed.

For m=1 and m=2, the positive derived subgroups are trivial. The consequence for the solvable filtration is stated for integers n≥1; it does not contradict the prior n=0 containment discussed in the original report.

The general signature, quasimorphism and amenable-group tools are credited prior results. The contribution presented is their application to this particular containment question, together with the explicit finite example. No historical firstness, exhaustive novelty clearance or continued worldwide openness is claimed.

For the separate explicit example, the supporting derivation reconstructs the Goeritz form from crossing geometry and records a printed index discrepancy in Erle’s matrix formula. A separate Burau–Meyer calculation gives the same signature. This qualification concerns the finite example; the arbitrary-depth existence proof is independent of that calculation.

Status and AI use

AI tools were used extensively in research, proof development, source checking, computation, writing and internal adversarial checking. AI-assisted reviews do not constitute independent human or journal peer review.

The manuscript and Zenodo edition are dated 15 October 2026 and identify version 1.0. The Zenodo record was created on the same date. These dates identify this edition; they do not establish first priority.

Read the paper

Open PDF in a full window · Download the original from Zenodo

Collaborate on this result

I welcome substantive collaboration from researchers in braid groups and low-dimensional topology on independent verification, comparison with existing work and development toward journal review. Coauthorship would reflect actual research contributions and mutual agreement.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). Deep derived pure braids need not be 1-solvable (Version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23393596

Download BibTeX

Background and prior work

• Harvey, with Park and Ray — Pure braids, Whitney towers, and 0-solvability: Question 3 on page 3021 states the derived-series containment question. The workshop report is numbered 2017/50 and was published in 2018.
• Gambaudo and Ghys — Braids and Signatures: Supplies the braid-closure signature quasimorphism, its homogenization and the signature values used in the general argument.
• Calegari — scl: Proposition 2.65 gives the rigidity of homogeneous quasimorphisms on amenable groups.
• Kawasaki, Kimura, Maruyama, Matsushita and Mimura — The space of nonextendable quasimorphisms: Lemma 7.12 gives the stronger prior result that restricting to a normal subgroup with solvable quotient preserves the quasimorphism defect.
• Otto — The (n)-solvable filtration of link concordance and Milnor’s invariants: Definition 2.3 fixes the integral smooth solution and standard-closure conventions.
• Toffoli — The Atiyah–Patodi–Singer rho invariant and signatures of links: Formula (1.1), Theorem 4.24 and Remark 4.25, equation (4.21), supply the bounding and compatible-framing surgery formulas used pointwise at −1 for the integral smooth solvability obstruction.
• Erle — Calculation of the signature of a 3-braid link: The checkerboard and correction conventions inform the separate finite example. The package records a printed matrix-index discrepancy and uses crossing-based reconstruction, corroborated by a separate Burau–Meyer signature calculation.