Please look at FLAT-VIRTUAL LINKS(which are obtained from knots in full torus) and try to think how to UPGRADE your invariants if we have ADDITIONAL FLAT CROSSINGS

https://arxiv.org/abs/2203.11040https://arxiv.org/abs/2210.09689

Does the invariant depend on the choice of the symplectic basis you chose at the beginning?

I see. Thank you very much!