Research ArticleNo Access

Arrow diagrams on spherical curves and computations

Noboru Ito

https://orcid.org/0000-0001-9011-8982

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo 153-8914, Japan

E-mail Address: nito@gm.ibaraki-ct.ac.jp

Current address: National Insititute of Technology, Ibaraki College, 866 Nakane, Hitachinaka, Ibaraki 312-8508, Japan.

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and

Masashi Takamura

School of Social Informatics, Aoyama Gakuin University, 5-10-1 Fuchinobe Chuo-ku, Sagamihara-shi, Kanagawa-ken 252-5258, Japan

E-mail Address: takamura@si.aoyama.ac.jp

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https://doi.org/10.1142/S0218216521500450Cited by:2 (Source: Crossref)

Abstract

We give a definition of an integer-valued function $\sum_{i} α_{i} x_{i}^{*}$ derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free $ℤ$ -module generated by the arrow diagrams with at most $l$ arrows, called relators of Type ( $\overset{̌}{I}$ ) (( $\overset{̌}{SII}$ ), ( $\overset{̌}{WII}$ ), ( $\overset{̌}{SIII}$ ) or ( $\overset{̌}{WIII}$ ), respectively), and introduce another function $\sum_{i} α_{i} {\tilde{x}}_{i}^{*}$ to obtain $\sum_{i} α_{i} x_{i}^{*}$ . One of the main results shows that if $\sum_{i} α_{i} {\tilde{x}}_{i}^{*}$ vanishes on finitely many relators of Type ( $\overset{̌}{I}$ ) (( $\overset{̌}{SII}$ ), ( $\overset{̌}{WII}$ ), ( $\overset{̌}{SIII}$ ) or ( $\overset{̌}{WIII}$ ), respectively), then $\sum_{i} α_{i} x_{i}^{*}$ is invariant under the deformation of type RI (strong RII, weak RII, strong RIII or weak RIII, respectively). The other main result is that we obtain new functions of arrow diagrams with up to six arrows explicitly. This computation is done with the aid of computers.

Keywords:

AMSC: 57K10, 57K16

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