If I take the HOMFLY(PT) polynomial defined by $$l \,P(L_+) + l^{-1}\,P(L_-) + m\,P(L_0) = 0,$$ I have looked at expressions of the form

(knots that are the same except inside a small disk, where they look like the pictures indicated).

In every case the result had a factor $(l^4 + 2l^2+1-l^2m^2)$.

My question is: why does the expression always have a factor $(l^4 + 2l^2+1-l^2m^2)$?

I understand that this happens when one of the links are disjoint due to the HOMFLY relation relating the disjoint sum (split union) and the connected sum: $$P(L_1 \sqcup L_2)=-\frac{l+l^{-1}}{m} P(L_1 \# L_2),$$ since if you stick this in, you get exactly the factor $(l^4 + 2l^2+1-l^2m^2)$.

Does the relation perhaps hold in general? According to the proof for the connected sum formula, it shouldn't.