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On the Kauffman polynomial of an adequate link. (English) Zbl 0645.57007
Using the main result, the author proves that a number of properties of a link follow from the existence of an “adequate” n-crossing diagram of the link. For example: If a link, L, has an adequate diagram with n- crossings then L has no projection with fewer crossings; and any two adequate diagrams of L have the same writhe.
The main result is a pair of formulae connecting a specialization of the Kauffman polynomial of a link diagram \(\Lambda (a,z)=\sum u_{r,s}a\) rz s, with the Tutte polynomials \(\chi\) of graphs \(G_+\), \(G_-\), \(\bar G{}_+\), \(\bar G{}_-\), associated with the graph of the link diagram. Set \(\phi \quad +(t)=\sum_{i}u_{i,n-i}t\quad i,\quad \phi \quad - (t)=\sum_{i}u_{-i,n-i}t\quad i\) \((u_{r,s}\) as above). Let \(\chi_ G(x,y)\) denote the Tutte polynomial of the graph G. Then the author proves the following: \(\phi \quad +(t)=\chi_{G_+}(0,t)\chi_{\bar G_-}(t,0)\) and \(\phi \quad -(t)=\chi_{G_-}(0,t)\chi_{\bar G_+}(t,0)\).
Reviewer: L.Neuwirth

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M15 Relations of low-dimensional topology with graph theory
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