TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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.

The Lawrence-Krammer-Bigelow representation is faithful

Statement

Assume AC. The representation $ρ_{L K B}$ of The Lawrence-Krammer-Bigelow representation is faithful for every $n \geq 1$ : if a boundary-fixed homeomorphism $σ$ represents an element of the kernel, then $σ$ is isotopic relative to $\partial D \cup P$ to $(Δ^{2})^{k}$ for some $k \in Z$ . For $n \geq 2$ the scalar value $q^{2 n} t^{2}$ forces $k = 0$ ; for $n = 1$ , $B_{1}$ is trivial.

Facts & Assumptions

Given: the standard disk with punctures $p_{1} < \dots < p_{n}$ and standard edges $E_{1}, \dots, E_{n - 1}$ ; a homeomorphism $σ$ of $(D, P)$ fixing $\partial D$ pointwise and representing an element of $\ker ρ_{L K B}$ .

[F1]

An LKB kernel braid fixes every standard adjacent edge up to isotopy: $σ (E_{i})$ is isotopic to $E_{i}$ relative to $\partial D \cup P$ for every $i$ , and $σ$ preserves the labelling of the punctures.

[F2]

A mapping class fixing all standard adjacent edges is a boundary twist power: a label-preserving boundary-fixed mapping class fixing each $E_{i}$ up to isotopy relative to $\partial D \cup P$ is isotopic to $(Δ^{2})^{k}$ for some $k \in Z$ .

[F3]

The full boundary twist acts on LKB by the scalar q to two n t squared: $ρ_{L K B} (Δ^{2 k}) = q^{2 n k} t^{2 k} id$ , and for $n \geq 2$ this scalar acts as the identity only for $k = 0$ .

[F4]

For $n = 1$ the group $B_{1}$ is trivial by the Artin presentation, so every representation of $B_{1}$ is faithful.

Proof

1.1F1F2F3givenalgebra

Assume $n \geq 2$ and let $[σ] \in \ker ρ_{L K B}$ . By [F1] the representative $σ$ preserves each marked label and fixes each standard edge up to isotopy, so [F2] produces $k \in Z$ with $σ$ isotopic relative to $\partial D \cup P$ to $(Δ^{2})^{k}$ . Since $ρ_{L K B}$ only depends on the isotopy class relative to $\partial D \cup P$ , $ρ_{L K B} (σ) = ρ_{L K B} (Δ^{2 k}) = q^{2 n k} t^{2 k} id$ by [F3].

2.1F3F4step 1.1algebra∎

Since $σ$ lies in the kernel, the scalar in step 1.1 is the identity; by [F3] this forces $k = 0$ . Hence $σ$ is isotopic relative to $\partial D \cup P$ to the identity, so it represents the trivial element of $B_{n}$ . Therefore $\ker ρ_{L K B}$ is trivial and $ρ_{L K B}$ is faithful for $n \geq 2$ . For $n = 1$ the claim is immediate by [F4].

Depends on

Used by

Every classical braid group is linear Corollary

Dependency tree · two levels

20 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

Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001) 471-486 (standard reference, not scraped)
Krammer, Braid groups are linear, Ann. of Math. 155 (2002) 131-156 (standard reference, not scraped)