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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 $$\phi \quad -(t)=\chi_{G_-}(0,t)\chi_{\bar G_+}(t,0)$$.
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M15 Relations of low-dimensional topology with graph theory