On invariants of graphs with applications to knot theory

Murasugi, Kunio

doi:10.1090/S0002-9947-1989-0930077-6

On invariants of graphs with applications to knot theory
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by Kunio Murasugi

Trans. Amer. Math. Soc. 314 (1989), 1-49

DOI: https://doi.org/10.1090/S0002-9947-1989-0930077-6

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Abstract:

To each weighted graph $\Gamma$, two invariants, a polynomial ${P_\Gamma }(x,y,z)$ and the signature $\sigma (\Gamma )$, are defined. The various partial degress of ${P_\Gamma }(x,y,z)$ and $\sigma (\Gamma )$ are expressed in terms of maximal spanning graphs of $\Gamma$. Furthermore, one unexpected property of Tutte’s dichromate is proved. These results are applied to knots or links in ${S^3}$.

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