Question 1 — If R = Q(q1,q 2), is the above map from Bn toHn(q1,q 2) injective?

Definitions and mathematical statement

Let $R={\mathbb{Q}}(q_1,q_2)$, the rational function field in the source's parameters, interpreted as independent indeterminates. Let $B_n$ be the $n$-strand braid group and retain its previously defined map $\rho_n:B_n\to H_n(q_1,q_2)$. The target algebra and "above map" are not defined in the supplied excerpt.

Question 1. Is that map injective over this generic coefficient field? For its intended values of $n$, the question is \[ \rho_n(\beta)=1\ \Longrightarrow\ \beta=1 \qquad(\beta\in B_n). \] The range of $n$ is not explicitly stated, and no specialization of $q_1,q_2$ is imposed.

If the missing setup is the standard two-parameter Hecke-algebra convention, \[ H_n(q_1,q_2)=R[B_n]/ \left\langle(\sigma_i-q_1)(\sigma_i-q_2):1\leq i<n\right\rangle, \] then $\rho_n$ is the canonical group map into the multiplicative units of this algebra. This displayed convention is conditional: different parameter or generator conventions must be checked against the source. The desired group injectivity is not injectivity of the group-algebra quotient $R[B_n]\to H_n$, and the target is not silently replaced by a Temperley--Lieb or another quotient algebra.

Short English statement

Is the source's braid-group map to the generic two-parameter algebra $H_n(q_1,q_2)$ faithful over ${\mathbb{Q}}(q_1,q_2)$? The algebra and map are omitted, so the usual Hecke interpretation is stated only conditionally.

Sources

• Local UnsolvedMath registry, permanent ID 11000258, source code catalogue entry, Question 1, PDF page 312. No individual proposer is supplied.
• Farb, Problems on Mapping Class Groups and Related Topics. Supplied URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf; recorded access 2026-07-29; extraction pdf-font-structure. The volume contains 329 explicitly typeset headings across chapters, with repeated numbering disambiguated by page.
• The summary mentions Hecke/Temperley--Lieb-type braid representations and an unexplained "deformation LL" connection, but gives no precise citation or verified theorem for this map.
• Supplied status open and OPEN-TRIAGE. The summary cannot verify the exact injectivity question. Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.