### Abstract

A chord diagram is a circle with paired points with each pair of points connected by a chord. Every generic immersed spherical curve pro- vides a chord diagram by associating each chord with two preimages of a double point. Any two spherical curves can be related by a finite sequence of three types of local replacement RI, RII, and RIII, called Reidemeister moves. This study counts the difference in the numbers of sub-chord dia- grams embedded in a full chord diagram of any spherical curve by applying one of the moves RI, strong RII, weak RII, strong RIII, and weak RIII defined by connections of branches related to the local replacements (Theorem 1.1). This yields a new integer-valued invariant under RI and strong RIII that provides a complete classification of prime reduced spherical curves with up to at least seven double points (Theorem 1.2, Fig. 24): there has been no such invariant before. The invariant expresses the necessary and sufficient condition that spherical curves can be related to a simple closed curve by a finite sequence consisting of RI and strong RIII (Theorem 1.3). Moreover, in- variants of spherical curves under ypes are provided by counting sub-chord diagrams (Theorem 1.4).

Original language | English |
---|---|

Pages (from-to) | 701-725 |

Number of pages | 25 |

Journal | Houston Journal of Mathematics |

Volume | 41 |

Issue number | 2 |

Publication status | Published - 2015 |

Externally published | Yes |

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### Keywords

- Chord diagrams
- Knot projections
- Reidemeister moves
- Spherical curves

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Houston Journal of Mathematics*,

*41*(2), 701-725.

**Sub-chord diagrams of knot projections.** / Ito, Noboru; Takimura, Yusuke.

Research output: Contribution to journal › Article

*Houston Journal of Mathematics*, vol. 41, no. 2, pp. 701-725.

}

TY - JOUR

T1 - Sub-chord diagrams of knot projections

AU - Ito, Noboru

AU - Takimura, Yusuke

PY - 2015

Y1 - 2015

N2 - A chord diagram is a circle with paired points with each pair of points connected by a chord. Every generic immersed spherical curve pro- vides a chord diagram by associating each chord with two preimages of a double point. Any two spherical curves can be related by a finite sequence of three types of local replacement RI, RII, and RIII, called Reidemeister moves. This study counts the difference in the numbers of sub-chord dia- grams embedded in a full chord diagram of any spherical curve by applying one of the moves RI, strong RII, weak RII, strong RIII, and weak RIII defined by connections of branches related to the local replacements (Theorem 1.1). This yields a new integer-valued invariant under RI and strong RIII that provides a complete classification of prime reduced spherical curves with up to at least seven double points (Theorem 1.2, Fig. 24): there has been no such invariant before. The invariant expresses the necessary and sufficient condition that spherical curves can be related to a simple closed curve by a finite sequence consisting of RI and strong RIII (Theorem 1.3). Moreover, in- variants of spherical curves under ypes are provided by counting sub-chord diagrams (Theorem 1.4).

AB - A chord diagram is a circle with paired points with each pair of points connected by a chord. Every generic immersed spherical curve pro- vides a chord diagram by associating each chord with two preimages of a double point. Any two spherical curves can be related by a finite sequence of three types of local replacement RI, RII, and RIII, called Reidemeister moves. This study counts the difference in the numbers of sub-chord dia- grams embedded in a full chord diagram of any spherical curve by applying one of the moves RI, strong RII, weak RII, strong RIII, and weak RIII defined by connections of branches related to the local replacements (Theorem 1.1). This yields a new integer-valued invariant under RI and strong RIII that provides a complete classification of prime reduced spherical curves with up to at least seven double points (Theorem 1.2, Fig. 24): there has been no such invariant before. The invariant expresses the necessary and sufficient condition that spherical curves can be related to a simple closed curve by a finite sequence consisting of RI and strong RIII (Theorem 1.3). Moreover, in- variants of spherical curves under ypes are provided by counting sub-chord diagrams (Theorem 1.4).

KW - Chord diagrams

KW - Knot projections

KW - Reidemeister moves

KW - Spherical curves

UR - http://www.scopus.com/inward/record.url?scp=84938372452&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84938372452&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84938372452

VL - 41

SP - 701

EP - 725

JO - Houston Journal of Mathematics

JF - Houston Journal of Mathematics

SN - 0362-1588

IS - 2

ER -