On the skein theory of dichromatic links and invariants of finite type

In [Dichromatic link invariants, Trans. Amer. Math. Soc. 321(1) (1990) 197-229], Hoste and Kidwell investigated the skein theory of oriented dichromatic links in S-3. They introduced a multi-variable polynomial invariant W. We use special substitutions for some of the parameters of the invariant W to show how to deduce invariants of finite type from W using partial derivatives. Then we consider the 2-component 1-trivial dichromatic links. We study the Vassiliev invariants of the 2-component in the complement of the 1-component, which is equivalent to studying Vassiliev invariants for knots in S-1 x D-2. We give combinatorial formulas for the type-zero and type-one invariants and we connect these invariants to existing invariants such as Aicardi's invariant. This provides us with a topological meaning of the first partial derivative, which is also shown to be universal as a type-one invariant.