Dichromatic link invariants

Hoste, Jim; Kidwell, Mark E.

doi:10.1090/S0002-9947-1990-0961623-2

Dichromatic link invariants
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by Jim Hoste and Mark E. Kidwell PDF

Trans. Amer. Math. Soc. 321 (1990), 197-229 Request permission

Abstract:

We investigate the skein theory of oriented dichromatic links in ${S^3}$. We define a new chromatic skein invariant for a special class of dichromatic links. This invariant generalizes both the two-variable Alexander polynomial and the twisted Alexander polynomial. Alternatively, one may view this new invariant as an invariant of oriented monochromatic links in ${S^1} \times {D^2}$, and as such it is the exact analog of the twisted Alexander polynomial. We discuss basic properties of this new invariant and applications to link interchangeability. For the full class of dichromatic links we show that there does not exist a chromatic skein invariant which is a mutual extension of both the two-variable Alexander polynomial and the twisted Alexander polynomial.

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