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

Khaled Bataineh

Jordan University of Science and Technology, Ar Ramtha, Irbid, Jordan

https://doi.org/10.1142/S0218216517500924Cited by:4 (Source: Crossref)

Abstract

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 $S^{1} \times D^{2} .$ 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.

Keywords:

AMSC: 57M27

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