The Kauffman bracket of virtual links and the Bollob\'as-Riordan   polynomial
  We show that the Kauffman bracket $[L]$ of a checkerboard colorable virtual link $L$ is an evaluation of the Bollob\'as-Riordan polynomial $R_{G_L}$ of a ribbon graph associated with $L$. This result generalizes Thistlethwaite's celebrated theorem relating the Kauffman bracket with the Tutte polynomial of planar graphs. 
Sergei Chmutov, Igor Pak