CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

An arbitrary isotopy of arcs need not be a braid isotopy

Statement refuted

Refuted claim: the requirement in Braid isotopy relative to the top and bottom endpoints that every slice of the family be a braid is redundant, that is: if $(α_{s})_{s \in I}$ is a continuous family of injective parameterisations of arcs in $D^{\circ} \times I$ whose endpoints $α_{s} (0)$ and $α_{s} (1)$ are held fixed at the bottom and top points of the base configuration throughout, then the images $A_{s} : = α_{s} [I]$ are strands of geometric braids, so that the family is a braid isotopy as soon as its two boundary arcs are braids.

The witness deforms the single strand of the trivial one-strand braid. At the middle of the deformation its height coordinate has a horizontal shelf: two distinct points at one and the same height, so its image meets a horizontal slice in two points and is therefore not the strand of any braid. Since reparametrising an arc does not change its image, no choice of parameterisation repairs this; the family is a legitimate continuous deformation of arcs with fixed endpoints, but it is not a braid isotopy, and the definition's one-point-per-height condition is not redundant.

What is and is not claimed. The two boundary arcs $A_{0}$ and $A_{1}$ of the exhibited family are equal (both are the trivial braid), so nothing here asserts that two braids fail to be braid-isotopic; what is refuted is only the claim that an arbitrary arc deformation with braid boundary arcs is itself a braid isotopy. Nothing is claimed about closed links or about the classification of knots. The height function of the middle arc is not strictly increasing; the definition of braid isotopy requires each intermediate object to be a braid, which for a single strand means exactly that it meets each horizontal slice once.

Facts & Assumptions

Given: The natural number $1$ , so that $h = \frac{1}{4 (1 + 1)} = \frac{1}{8}$ and the base configuration of Geometric braids in the disc with setwise endpoints is $Q = (q_{1})$ with $q_{1} = (0, 0)$ , and the family of arc parameterisations $α_{s} ⁣ : I \to D^{\circ} \times I$ , $s \in I$ , defined by

$λ (s) : = \frac{1}{4} \min (2 s, 2 - 2 s), w (u) : = {\begin{cases} 4 u, & 0 \leq u \leq \frac{1}{4}, \\ 2 - 4 u, & \frac{1}{4} \leq u \leq \frac{3}{4}, \\ 4 u - 4, & \frac{3}{4} \leq u \leq 1, \end{cases} α_{s} (u) : = ((\frac{λ (s)}{8} w (u), 0), u + λ (s) w (u)) .$

[F1]

For $n = 1$ a braid based at $Q$ is a continuous map $z_{1} ⁣ : I \to D^{\circ}$ with $z_{1} (0) = q_{1}$ and ${z_{1} (1)} = {q_{1}}$ , and its strand is the graph ${(z_{1} (t), t) : t \in I} \subseteq D^{\circ} \times I$ , which by construction meets each horizontal slice $D^{\circ} \times {t}$ , $t \in I$ , in exactly one point; the trivial braid has the constant strand $z_{1} (t) = q_{1}$ , whose graph is ${(0, 0)} \times I$ , and a braid isotopy from $β$ to $β^{'}$ is a family of braids $Z (s, \cdot)$ depending jointly continuously on $s$ , with each slice a braid (Geometric braids in the disc with setwise endpoints, Braid isotopy relative to the top and bottom endpoints, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[F2]

The closed unit disc is $D = {w \in R^{2} : ∥ w ∥_{2} \leq 1}$ with interior $D^{\circ}$ ; the set $D^{\circ} \times I$ carries the product topology, and its points are written as $(spatial point, height)$ (Euclidean spheres and closed balls as subspaces of $R^{n}$ , The product set $\prod_{i \in I} X_{i}$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space).

[F3]

The functions $s \mapsto λ (s)$ and $u \mapsto w (u)$ are continuous, being piecewise linear with continuous gluing, and sums, products and composites of continuous maps are continuous; continuity of a function on $I$ or on $I \times I$ follows from continuity on the finitely many closed pieces ${s \leq \frac{1}{2}}$ , ${s \geq \frac{1}{2}}$ and ${u \leq \frac{1}{4}}$ , ${\frac{1}{4} \leq u \leq \frac{3}{4}}$ , ${u \geq \frac{3}{4}}$ (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous, Continuity of a map of topological spaces at a point and globally, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

Counterexample

technique · direct

1.1

The family is jointly continuous with the prescribed fixed endpoints. By [F3] the maps $s \mapsto λ (s)$ and $u \mapsto w (u)$ are continuous, hence $(s, u) \mapsto α_{s} (u)$ is continuous by [F3]; at $u = 0$ one has $w (0) = 0$ , so $α_{s} (0) = ((0, 0), 0) = (q_{1}, 0)$ for every $s$ , and at $u = 1$ one has $w (1) = 0$ and $λ (s) w (1) = 0$ , so $α_{s} (1) = ((0, 0), 1) = (q_{1}, 1)$ for every $s$ ; thus the endpoints of the parameterised arcs are fixed at the bottom and top points of $Q$ .

F1F2F3

1.2

Every slice lies in $D^{\circ} \times I$ . For every $s \in I$ one has $0 \leq λ (s) \leq \frac{1}{4}$ , and $w$ takes the values $w (0) = 0$ , $w (\frac{1}{4}) = 1$ , $w (\frac{1}{2}) = 0$ , $w (\frac{3}{4}) = - 1$ , $w (1) = 0$ with $∣ w (u) ∣ \leq 1$ for all $u$ ; hence the spatial coordinate satisfies $∣ \frac{λ (s)}{8} w (u) ∣ \leq \frac{1}{32} < 1$ , so each spatial point $(\frac{λ (s)}{8} w (u), 0)$ lies in $D^{\circ}$ ; and the height $u + λ (s) w (u)$ lies in $I$ , because $w (u) \geq 0$ for $u \leq \frac{1}{2}$ gives $0 \leq u \leq u + λ (s) w (u) \leq \frac{1}{2} + \frac{1}{4} \leq 1$ there, while $w (u) \leq 0$ for $u \geq \frac{1}{2}$ gives $\frac{1}{2} - \frac{1}{4} \leq u + λ (s) w (u) \leq u \leq 1$ there.

F2F3

2.1

Each slice is a simple arc. Let $s \in I$ and suppose $α_{s} (u) = α_{s} (u^{'})$ ; comparing the spatial coordinates gives $λ (s) w (u) = λ (s) w (u^{'})$ and comparing the heights gives $u - u^{'} = λ (s) (w (u^{'}) - w (u)) = 0$ , so $u = u^{'}$ ; hence every $α_{s}$ is injective, and its image $A_{s} : = α_{s} [I]$ is a simple arc with the endpoints $(q_{1}, 0)$ and $(q_{1}, 1)$ .

step 1.1step 1.2

2.2

The middle slice is not the strand of a braid. At $s = \frac{1}{2}$ one has $λ (\frac{1}{2}) = \frac{1}{4}$ , and the two parameters $u = \frac{1}{4}$ and $u = \frac{3}{4}$ are distinct while $w (\frac{1}{4}) = 1$ , $w (\frac{3}{4}) = - 1$ give equal heights $\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$ and $\frac{3}{4} - \frac{1}{4} = \frac{1}{2}$ and spatial coordinates $\frac{1}{32}$ and $- \frac{1}{32}$ ; hence $α_{1 / 2} (\frac{1}{4}) = ((\frac{1}{32}, 0), \frac{1}{2})$ and $α_{1 / 2} (\frac{3}{4}) = ((- \frac{1}{32}, 0), \frac{1}{2})$ are two distinct points of $A_{1 / 2}$ in the same horizontal slice $D^{\circ} \times {\frac{1}{2}}$ .

F2step 1.2

3.1

The boundary arcs are braids. For $s \in {0, 1}$ one has $λ (s) = \frac{1}{4} \min (0, 2) = 0$ , so $α_{s} (u) = ((0, 0), u)$ ; the image $A_{0} = A_{1} = {(0, 0)} \times I$ is exactly the strand of the trivial braid of [F1], hence both boundary slices are geometric braids based at $Q$ and the hypothesis of the refuted claim is satisfied.

F1step 1.1step 2.1

4.1

Conclusion. The family $(α_{s})$ satisfies the hypotheses of the refuted claim by steps 1.1, 1.2, 2.1 and 3.1, but its slice at $s = \frac{1}{2}$ meets the horizontal slice at height $\frac{1}{2}$ in two distinct points by step 2.2, whereas the strand of a braid based at $Q$ meets each horizontal slice in exactly one point by [F1]; since a reparametrisation does not change the image $A_{1 / 2}$ , no choice of parameterisation makes that slice a braid, so the family is not a braid isotopy. Hence the one-point-per-height requirement in the definition of braid isotopy is not redundant, and the refuted claim is false. ∎

F1step 2.1step 3.1step 2.2

Remarks

At $s = 1 / 2$ the height function is $2 u$ on $[0, 1 / 4]$ , constant $1 / 2$ on $[1 / 4, 3 / 4]$ , and $2 u - 1$ on $[3 / 4, 1]$ . This horizontal shelf gives many distinct points at height $1 / 2$ ; the height is nondecreasing, but not strictly increasing, and the arc is not a one-point-per-height graph.
The example is one-dimensional in the sense that a single strand suffices: no collision analysis between different strands arises, and the whole phenomenon is the failure of the height projection to restrict to a homeomorphism of the arc onto $I$ .
The two boundary arcs being equal is what makes the point sharp: the deformation does not change the isotopy class of anything, and yet it leaves the class of braid isotopies, because braid isotopy is a relation between braids (tuples of one-point-per-height strands) and not between arcs.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

47 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Juan Gonzalez-Meneses, Basic results on braid groups, sections 1.2-1.3, printed pp. 4-5 (standard reference, not scraped)
Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 1.1, author manuscript pp. 3-5 (standard reference, not scraped)