On the definition of graph index (Q2861590)
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scientific article; zbMATH DE number 6224471
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the definition of graph index |
scientific article; zbMATH DE number 6224471 |
Statements
11 November 2013
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skein polynomial
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graph index
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braid index
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Bennequin surface
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On the definition of graph index (English)
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In their memoir [``An index of a graph with applications to knot theory'', Mem. Am. Math. Soc. 508, 101 p. (1993; Zbl 0792.05047)], \textit{K. Murasugi} and \textit{J. H. Przytycki} obtained the inequality NEWLINE\[NEWLINEb(L)\leq s(D)-\text{ind}(D)NEWLINE\]NEWLINE where \(D\) is a diagram of \(s(D)\) Seifert circles of a link \(L\) with braid index \(b(L)\), and \(\text{ind}(D)\) is the index of the Seifert graph of \(D\). The author points out that the general case in the Murasugi-Przytycki paper missed some cases. This led to a modification of \(\text{ind}(D)\) which the author calls \(\text{ind}_0(D)\). It is then shown that this modification provides the desired result as Murasugi and Przytycki intended.
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