Sub-chord diagrams of knot projections

Noboru Ito, Yusuke Takimura

Research output: Contribution to journalArticle

3 Citations (Scopus)

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 languageEnglish
Pages (from-to)701-725
Number of pages25
JournalHouston Journal of Mathematics
Volume41
Issue number2
Publication statusPublished - 2015
Externally publishedYes

Fingerprint

Chord Diagrams
Knot
Projection
Curve
Chord or secant line
Replacement
Invariant
Theorem
Simple Closed Curve
Strong Theorems
Counting
Count
Circle
Branch
Express
Necessary Conditions
Integer
Sufficient Conditions

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Ito, N., & Takimura, Y. (2015). Sub-chord diagrams of knot projections. Houston Journal of Mathematics, 41(2), 701-725.

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

In: Houston Journal of Mathematics, Vol. 41, No. 2, 2015, p. 701-725.

Research output: Contribution to journalArticle

Ito, N & Takimura, Y 2015, 'Sub-chord diagrams of knot projections', Houston Journal of Mathematics, vol. 41, no. 2, pp. 701-725.
Ito, Noboru ; Takimura, Yusuke. / Sub-chord diagrams of knot projections. In: Houston Journal of Mathematics. 2015 ; Vol. 41, No. 2. pp. 701-725.
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