Question 6 — DoesX3 equal 0 in Zn?

Definitions and mathematical statement

The algebraic setup is supplied in the preceding local record 11000262. Let $B_n$ be the braid group and $RB_n$ its group algebra over the source's coefficient ring $R$, with parameter $q$. Write $\sigma_{i_1\cdots i_k}=\sigma_{i_1}\cdots\sigma_{i_k}$, and retain the source's barred-word notation $\bar\sigma_{i_1\cdots i_k}$ for its inverse-word convention. The extraction does not clearly distinguish inversion of the whole word from replacing letters by inverses in the written order; this matters for words of length greater than one.

The source defines \[ X_2=q\sigma_1^{-1}+(1-q)1-\sigma_1,\qquad X_k=(q^{k-1}\bar\sigma_{(k-1)\cdots1}-\sigma_{1\cdots(k-1)})X_{k-1}\quad(k\ge3). \] It defines $Z_n$ as a quotient of $RB_n$ by the relations beginning with \[ (q\sigma_2^{-1}+1-q-\sigma_2)X_2 =(q\bar\sigma_{21}-\sigma_{12})X_2, \] and continuing in the pattern \[ (q^{j-1}\bar\sigma_{j\cdots2}-\sigma_{2\cdots j})X_j =(q^{j-1}\bar\sigma_{j\cdots1}-\sigma_{1\cdots j})X_j. \] The written list ends with $j=n$. Under the standard $B_n$ convention only $\sigma_1,\ldots,\sigma_{n-1}$ exist, so this endpoint requires correction or a different braid indexing convention. No endpoint correction is silently chosen here. The ring $R$ and its parameters also are not specified in the extracted setup.

Question 6. Does the element $X_3$ vanish in $Z_n$? Once the coefficient, inverse-word, and index conventions are fixed, this is the ideal-membership question \[ X_3\in I_n\subset RB_n, \] where $I_n$ is the two-sided ideal generated by the defining relations; $n$ must be in a range where $X_3$ and the relations are defined. It is not asking whether an independent generator named $X_3$ may be declared zero.

The source suggests that additional relations may be needed, for example \[ \sigma_1X_2=tX_2, \] or a more general relation. The parameter $t$ and its relation to $R$ require specification. This suggestion is not already imposed in the question. The preceding record says the Iwahori--Hecke algebra $H_n(1,-q)$ is obtained by imposing $X_2=0$, and the Birman--Wenzl--Murakami quotient by imposing both $X_3=0$ and $\sigma_1X_2=tX_2$. Thus the question probes whether the proposed $Z_n$ genuinely retains more information than that quotient; a quotient relation cannot itself prove $X_3=0$ in the original algebra.

Short English statement

Determine whether $X_3$ is already zero in Bigelow's proposed braid-group algebra quotient, before adding the relations that recover the Birman--Wenzl--Murakami algebra. The extracted definition has inverse-word, coefficient-ring, and generator-index ambiguities.

Sources

• Local UnsolvedMath registry, permanent ID 11000263, source code catalogue entry; no separate proposer or proposal year supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics, Question 6, PDF page 323, in source text bearing S. Bigelow's chapter attribution; https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The registry records PDF-font-structure extraction and access on 2026-07-29; its inventory distinguishes 329 labeled headings by source page.
• Adjacent local ID 11000262 contains the $X_k$ recursion, $Z_n$ relations, and the Hecke/Birman--Wenzl--Murakami quotient descriptions. Its homological motivation is expressly described by the source as unclear and unpublished at that time.