TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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 Khovanov-Seidel weak braid action is faithful

Statement

Assume AC, inherited from the braid-to-mapping-class dictionary used in the Hom theorem and the detector, and from the supplied representative-independence and isotopy invariance of intersection numbers. For every $m \geq 1$ the weak action of $B_{m + 1}$ on $C_{m}$ given by the complexes $R_{σ}$ (The Khovanov-Seidel complexes give a weak derived braid action) is faithful (Faithful weak action): if $R_{σ} ≅ {Id}_{C_{m}}$ for a braid $σ$ , then $σ = 1$ . Equivalently, no nontrivial braid acts by the identity functor, although nontrivial braids may act trivially on the Grothendieck group $G (A_{m})$ .

Facts & Assumptions

Given: AC, the weak braid action by the complexes $R_{σ}$ on $C_{m}$ , the basic arcs $b_{0}, \dots, b_{m}$ with their normalized bigradings, and a braid $σ \in B_{m + 1}$ with $R_{σ} ≅ {Id}_{C_{m}}$ .

[L1]

For all $σ, τ$ and all $j, k, s_{1}, s_{2}$ the Hom groups ${Hom}_{C_{m}} (R_{τ} P_{k}, R_{σ} P_{j} [s_{1}] {- s_{2}})$ are free with Poincaré polynomial $I^{b i g r} ({\tilde{f}}_{τ} {\tilde{b}}_{k}, {\tilde{f}}_{σ} {\tilde{b}}_{j})$ (Homs compute bigraded arc intersections).

[L2]

If $f \in G$ satisfies $I (b_{j}, f (b_{k})) = I (b_{j}, f^{2} (b_{k})) = I (b_{j}, b_{k})$ for all $j, k$ , then $[f] = 1$ in $G$ (The basic arcs detect the identity braid).

[L3]

Under AC, the preferred lifts give the action of $B_{m + 1}$ on bigraded curves, and $I^{b i g r} ({\tilde{c}}_{0}, {\tilde{c}}_{1}) ∣_{q_{1} = q_{2} = 1} = 2 I (c_{0}, c_{1})$ , so equality of the bigraded numbers specializes to equality of the ordinary intersection numbers (Local indices and bigraded intersection numbers).

[L4]

Under the isomorphism $B_{m + 1} ≅ G = π_{0} Diff (D, \partial D; Δ)$ a braid $σ$ corresponds to a boundary-fixed mapping class $f_{σ}$ well defined up to isotopy, and $σ = 1$ iff $[f_{σ}] = 1$ (Basic arcs, admissible curves and the standard normal form, whose dictionary includes Artin-presentation completeness and the smooth comparison).

Proof

technique · direct

1.1L1

The Hom-table of $R_{σ}$ is the identity table. Suppose $R_{σ} ≅ {Id}_{C_{m}}$ . Then for all $j, k$ and all shifts $s_{1}, s_{2}$ the induced isomorphism gives ${Hom}_{C_{m}} (P_{k}, R_{σ} P_{j} [s_{1}] {- s_{2}}) ≅ {Hom}_{C_{m}} (P_{k}, P_{j} [s_{1}] {- s_{2}}) .$ By the Hom theorem [L1] applied with $τ = 1$ on the left and $σ = 1$ on the right, taking Poincaré polynomials gives $I^{b i g r} ({\tilde{b}}_{k}, σ {\tilde{b}}_{j}) = I^{b i g r} ({\tilde{b}}_{k}, {\tilde{b}}_{j}) for all j, k .$

2.1step 1.1L1

The same for the square. The weak-action relation $R_{σ^{2}} ≅ R_{σ} R_{σ}$ gives $R_{σ^{2}} ≅ {Id}_{C_{m}}$ as well; hence the same argument yields $I^{b i g r} ({\tilde{b}}_{k}, σ^{2} {\tilde{b}}_{j}) = I^{b i g r} ({\tilde{b}}_{k}, {\tilde{b}}_{j}) for all j, k .$

3.1step 1.1step 2.1L3L4

Specialization to the ordinary intersection table. Setting $q_{1} = q_{2} = 1$ in the identities of steps 1.1 and 2.1 and using [L3] gives $I (b_{k}, f_{σ} (b_{j})) = I (b_{k}, b_{j}), I (b_{k}, f_{σ}^{2} (b_{j})) = I (b_{k}, b_{j}) for all j, k,$ where $f_{σ}$ is the boundary-fixed mapping class of $σ$ [L4].

4.1step 3.1L2L4

The detector concludes. The two families of equalities of step 3.1 are exactly the hypotheses of the detector lemma [L2] for $f = f_{σ}$ ; hence $[f_{σ}] = 1$ in $G$ , and by [L4] $σ = 1$ in $B_{m + 1}$ . This proves faithfulness.

5.1step 4.1L1L2L3∎

Conclusion. No nontrivial braid acts by the identity functor: the identity of the Hom tables forces the identity of the mapping class, by the two-iterate hypothesis of the detector. The contrast with the Grothendieck group is displayed by the decategorification proposition and the companion counterexample. AC is inherited through the Hom theorem and the detector, which use $B_{m + 1} ≅ G$ , and through the supplied representative-independence and isotopy invariance of intersection numbers in [L3].

Depends on

Used by

Equal actions on K₀ do not imply isomorphic derived autoequivalences Counterexample

Dependency tree · two levels

50 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

Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, J. Amer. Math. Soc. 15 (2002) 203-271, Corollary 1.2 (standard reference, not scraped)