Homological invariants of links in a thickened surface

Abstract

In the paper we introduce a new invariant of oriented links in a thickened surface. The invariant is a generalization of polynomial invariants $$u_{\pm }(t)$$ introduced by Turaev in 2008. Our invariant $$Q(\ell )$$ is defined as follows. Consider a diagram representing an oriented link $$\ell \subset \varSigma \times [0,1]$$, where $$\varSigma $$ is a closed orientable surface of positive genus. A value of $$Q(\ell )$$ is the formal sum over all crossings in the diagram terms of the form $${\text {sign}}(c)[h_1(c),h_2(c)]$$, where $${\text {sign}}(c)$$ denotes the sign of a crossing c and $$[h_1(c),h_2(c)]$$ 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 $$u_{\pm }(t)$$ is weaker than $$Q(\ell )$$ and discuss the results of a computing experiment which illustrates the fact.