In the paper we introduce a new invariant of oriented links in a thickened surface. The invariant is a generalization of polynomial invariants u_±(t) introduced by Turaev in 2008. Our invariant Q(ℓ) is defined as follows. Consider a diagram representing an oriented link ℓ⊂ Σ× [0 , 1] , where Σ is a closed orientable surface of positive genus. A value of Q(ℓ) is the formal sum over all crossings in the diagram terms of the form sign (c) [h₁(c) , h₂(c)] , where sign (c) denotes the sign of a crossing c and [h₁(c) , h₂(c)] denotes a ordered pair of homology classes of two loops associated with the crossing. As an application we prove a low bounds for the crossing number and for the virtual genus of a link. Additionally we describe an analogous constructions in some other situations, in particular, in the case of long virtual knots and prove a low bound for the virtual genus of a virtual knot which can be represented by concatenation of long virtual knots. Finally we show that Turaev’s invariants u_±(t) is weaker than Q(ℓ) and discuss the results of a computing experiment which illustrates the fact.