The generalized one-layer conjecture for framed graph-layer presentations

Let

$$\Omega=T_{i_mj_m}^{k_m}\cdots T_{i_1j_1}^{k_1}$$

be a graph-layer word in the twists $T_{ij}$, and let $\mathbf m=(m_1,\ldots,m_g)\in\mathbb Z^g$. Put

$$V_{\mathbf m}=T_{x_1}^{m_1}\cdots T_{x_g}^{m_g}.$$

The associated framed graph-layer presentation is

$$G(\mathbf m,\Omega)=\left\langle a_1,\ldots,a_g\\ \middle|\\ \gamma V_{\mathbf m}\Omega(y_1),\ldots,\gamma V_{\mathbf m}\Omega(y_g)\right\rangle.$$

A framed graph-layer presentation is framed generalized one-layer when its underlying graph-layer word is equivalent, using the defining relations among the twists $T_{ij}$, to a recursively peelable word. Generalized one-layer conjecture. If a framed graph-layer presentation defines the trivial group, then it is a framed generalized one-layer presentation. The conjecture asserts that triviality forces a recursively peelable twist representative; its status and consequences for the proposed classification of these presentations remain open in the source.