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Let $\Sigma$ be an oriented surface. I want to know about link invariants in $\Sigma\times [0,1]$. I already know the Ozawa polynomial introduced in this paper, but I couldn’t find any other than that. Also I want to know open problems about them.

Any comments are welcome.

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    $\begingroup$ If you are interested in invariants that are preserved under stabilization & destabilization of $\Sigma$, then Carter, Kamada and Saito showed these are exactly the same thing as "virtual link invariants." See arxiv.org/abs/math/0008118 $\endgroup$
    – Eric S.
    Commented Jul 25 at 17:09
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    $\begingroup$ Kuperberg furthermore showed that "virtual links" are exactly the same thing as links on thickened surfaces that are irreducible with respect to destabilizing. See arxiv.org/abs/math/0208039 $\endgroup$
    – Eric S.
    Commented Jul 25 at 17:11

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