## Abstract

The universal order 1 invariant f^{U} of immersions of a closed orientable surface into R^{3}, 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 sumf^{U} = under(⊕, n ∈ Z) f_{n}^{H} ⊕ under(⊕, n ∈ Z) f_{n}^{T} ⊕ M ⊕ Q where each f_{n}^{H}, f_{n}^{T} 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 f_{n}^{H}, f_{n}^{T} on all immersions, and for the value of M on any embedding.

Original language | English |
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Pages (from-to) | 358-372 |

Number of pages | 15 |

Journal | Advances in Mathematics |

Volume | 206 |

Issue number | 2 |

DOIs | |

State | Published - 10 Nov 2006 |

### Bibliographical note

Funding Information:E-mail address: tahl@math.biu.ac.il. URL: http://www.math.biu.ac.il/~tahl. 1 Partially supported by the Minerva Foundation.

## Keywords

- Finite order invariants
- Immersions of surfaces