Conjectural higher Khovanov, Jones, and δ-polynomials for T(3,6n ± 1)

Let $\mathcal{S}^n=T(3,6n+1)$ and $\mathcal{T}^n=T(3,6n-1)$ for $n greater than or equal to 1$. Define the auxiliary polynomials $f_n(t,q)=sum_{k=0}^{n-1}t^8q^{12}$ and $f_n(q)=sum_{k=0}^{n-1}q^6$. Let $E^k_L(t,q)$, $V^k_L(q)$, and $U^k_L(delta)$ denote respectively the higher Khovanov, Jones, and $delta$-polynomials associated to the $E^k$ page, with $s$ and $sigma$ the grading shifts specified by $s(\mathcal{S}_n)=12n$, $s(\mathcal{T}_n)=12n-4$, and $sigma(\mathcal{S}_n)=sigma(\mathcal{T}_n)=8n$. Conjectural polynomial formulas. For every $n greater than or equal to 1$, these higher polynomials are given by the displayed formulas in the source, including the stated expressions for $E^2$, $E^3$, $E^4$, $V^2$, $V^3$, $V^4$, $U^2$, $U^3$, and $U^4$ for $\mathcal{S}^n$ and $\mathcal{T}^n$; in particular, both $U^4_{\mathcal{S}^n}(delta)$ and $U^4_{\mathcal{T}^n}(delta)$ have the stated alternating-power form. These formulas are presented as a conjectural data set for seeking a combinatorial description of the higher pages; their validity and a general derivation remain open.