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 = under(⊕, n ∈ Z) fnH ⊕ under(⊕, n ∈ Z) fnT ⊕ 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.
Bibliographical noteFunding Information:
E-mail address: email@example.com. URL: http://www.math.biu.ac.il/~tahl. 1 Partially supported by the Minerva Foundation.
- Finite order invariants
- Immersions of surfaces