Polynomial invariants for unoriented links and framed links

Let $T_n$ be a trivial link of $n$ components, and let $L_0,L_1,L_2,L_3$ be a standard unoriented skein quadruple, with $L_{i+1}$ obtained from $L_i$ by a right-handed half-twist on the two involved arcs. Polynomial-invariant conjecture. There is a polynomial invariant of unoriented links in $S^3$, $P_1(L)\in Z[x,t]$, satisfying

$$P_1(T_n)=t^n,$$

and

$$P_1(L_0)+xP_1(L_1)-xP_1(L_2)-P_1(L_3)=0.$$

There is also a polynomial invariant of unoriented framed links, $P_2(L)\in Z[A^{\pm1},t]$, satisfying

$$P_2(T_n)=t^n,$$ $$P_2(L^{(1)})=-A^3P_2(L),$$

and

$$P_2(L_0)+A(A^2+A^{-2})P_2(L_1)+(A^2+A^{-2})P_2(L_2)+AP_2(L_3)=0,$$

where $L^{(1)}$ is obtained from $L$ by a positive half-twist on its framing. These invariants are proposed in connection with possible linear independence of trivial links when $b_{\infty}=0$ and $b_0b_1=b_2b_3$; their existence remains open in the source.