ABSTRACT
This chapter goals to improve our weak version of the Kauffman-Murasugi-Thistlethwaite theorem. It utilizes the fact that virtual links are in one-to-one correspondence with knots in surfaces and our results apply to classical link diagrams in surfaces. A connected, classical link diagram without nugatory crossings is a reduced classical link diagram. A reduced, alternating classical diagram has the proper boundary property. A bound on the span of the f-polynomial for virtual link diagrams would: determine bounds on the crossing number and genus of a virtual knot and distinguish classical versus non-classical knots. For a virtual link diagram D, a state s of the diagram is determined by selecting a smoothing type for each classical crossing of the diagram. The chapter discusses the diagram by changing a single classical crossing into a virtual crossing.
