Formulae for order one invariants of immersions of surfaces

The universal order 1 invariant fU of immersions of a closed orientable surface into R3, whose existence has been established in [T. Nowik, Order one invariants of immersions of surfaces into 3-space, Math. Ann. 328 (2004) 261-283], is the direct sumfU=⊕n∈ZfnH⊕⊕n∈ZfnT⊕M⊕Q where each fnH,fnT is a Z valued invariant and M,Q are Z/2 valued invariants. An explicit formula for the value of Q on any embedding has been given in [T. Nowik, Automorphisms and embeddings of surfaces and quadruple points of regular homotopies, J. Differential Geom. 58 (2001) 421-455]. In the present work we give explicit formulae for the value of each fnH,fnT on all immersions, and for the value of M on any embedding.