Problem 6.3 — (S.J.

Definitions and mathematical statement

The six-strand Artin braid group is \[ B_6=\langle\sigma_1,\ldots,\sigma_5\mid \sigma_i\sigma_j=\sigma_j\sigma_i\ (|i-j|\geq2),\quad \sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}\ (1\leq i\leq4)\rangle. \] Let $\rho:B_6\to\operatorname{Aut}(V_2^6)$ be the action referred to in the source. Is it faithful, that is, \[ \ker\rho=\{1\}\,? \] Equivalently, must every nonidentity braid act nontrivially on this particular module?

The extract does not define $V_2^6$, its scalar ring/field, the generator operators, or any deformation parameter and specialization. These data can change the kernel and are indispensable; the notation alone does not specify a matrix representation. The question concerns the given full $B_6$ action, not a projectivization, a quotient by the center, or the existence of some other faithful braid representation.

Short English statement

Is the source's particular six-strand braid action on $V_2^6$ injective? The module, operators, coefficients, and parameter specialization are absent, so a different faithful braid representation would not answer it.

Sources

• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 6.3, PDF page 96, attributed to S. J. Bigelow, https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf; accessed 2026-07-29, PDF-font-structure extraction.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400111$, source code catalogue entry, Problem 6.3. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.