Type $C$ skein modules and transparent elements

We study skein modules using $Sp(2n)$ webs. We define multivariable analogues of Chebyshev polynomials in the Type $C$ setting and use them to construct transparent elements in the skein module at roots of unity. Our arguments are diagrammatic and make use of an explicit braiding formula for $1$ and $k$ labeled strands and an analogue of Kuperberg’s tetravalent vertex in the annular setting.