### Abstract

In this note, we define a notion of finite-type for invariants of curves on surfaces as an analogue of the notion of finite-type for invariants of knots and 3-manifolds (Section 3). We also present a systematic construction for a large family of finite-type invariants SCI_{n} for curves on surfaces (Section 5). Arnold's invariants of plane isotopy classes of plane curves occur as invariants of order 1. Our theory of finite-type invariants of curves on surfaces is developed using the topological theory of words.

Original language | English |
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Pages (from-to) | 129-134 |

Number of pages | 6 |

Journal | Proceedings of the Japan Academy Series A: Mathematical Sciences |

Volume | 85 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2009 Nov |

### Fingerprint

### Keywords

- Arnold's invariants
- Finite-type invariants
- Immersed curves
- Topological theory of words

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Proceedings of the Japan Academy Series A: Mathematical Sciences*,

*85*(9), 129-134. https://doi.org/10.3792/pjaa.85.129

**Finite-type invariants for curves on surfaces.** / Ito, Noboru.

Research output: Contribution to journal › Article

*Proceedings of the Japan Academy Series A: Mathematical Sciences*, vol. 85, no. 9, pp. 129-134. https://doi.org/10.3792/pjaa.85.129

}

TY - JOUR

T1 - Finite-type invariants for curves on surfaces

AU - Ito, Noboru

PY - 2009/11

Y1 - 2009/11

N2 - In this note, we define a notion of finite-type for invariants of curves on surfaces as an analogue of the notion of finite-type for invariants of knots and 3-manifolds (Section 3). We also present a systematic construction for a large family of finite-type invariants SCIn for curves on surfaces (Section 5). Arnold's invariants of plane isotopy classes of plane curves occur as invariants of order 1. Our theory of finite-type invariants of curves on surfaces is developed using the topological theory of words.

AB - In this note, we define a notion of finite-type for invariants of curves on surfaces as an analogue of the notion of finite-type for invariants of knots and 3-manifolds (Section 3). We also present a systematic construction for a large family of finite-type invariants SCIn for curves on surfaces (Section 5). Arnold's invariants of plane isotopy classes of plane curves occur as invariants of order 1. Our theory of finite-type invariants of curves on surfaces is developed using the topological theory of words.

KW - Arnold's invariants

KW - Finite-type invariants

KW - Immersed curves

KW - Topological theory of words

UR - http://www.scopus.com/inward/record.url?scp=77949338454&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77949338454&partnerID=8YFLogxK

U2 - 10.3792/pjaa.85.129

DO - 10.3792/pjaa.85.129

M3 - Article

AN - SCOPUS:77949338454

VL - 85

SP - 129

EP - 134

JO - Proceedings of the Japan Academy Series A: Mathematical Sciences

JF - Proceedings of the Japan Academy Series A: Mathematical Sciences

SN - 0386-2194

IS - 9

ER -