Research ArticleNo Access

Knot diagrams on a punctured sphere as a model of string figures

Graduate School of Fundamental Science and Engineering, Waseda University, Okubo 3-4-1, Shinjuku, Tokyo 169-8555, Japan (graduated)

E-mail Address: araimasa23@gmail.com

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and

Kouki Taniyama

https://orcid.org/0000-0002-4759-8543

Department of Mathematics, School of Education, Waseda University, Nishi-Waseda 1-6-1, Shinjuku-ku, Tokyo, 169-8050, Japan

E-mail Address: taniyama@waseda.jp

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https://doi.org/10.1142/S0218216520500716Cited by:1 (Source: Crossref)

Abstract

A string figure is topologically a trivial knot lying on an imaginary plane orthogonal to the fingers with some crossings. The fingers prevent cancellation of these crossings. As a mathematical model of string figure, we consider a knot diagram on the $x y$ -plane in $x y z$ -space missing some straight lines parallel to the $z$ -axis. These straight lines correspond to fingers. We study minimal number of crossings of these knot diagrams under Reidemeister moves missing these lines.

Keywords:

AMSC: Primary: 57K10, Secondly: 57K20

References

1. P. Cahn , A generalization of the Turaev cobracket and the minimal self-intersection number of a curve on a surface, New York J. Math. 19 (2013) 253–283. Web of Science, Google Scholar
2. J. Hass and P. Scott , Intersections of curves on surfaces, Israel J. Math. 51 (1985) 90–120. Crossref, Web of Science, Google Scholar
3. K. Tagami , A Khovanov type invariant derived from an unoriented HQFT for links in thickened surfaces, Internat. J. Math. 24 (2013) 1350078. Link, Web of Science, Google Scholar
4. V. Turaev , Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. École Norm. Sup. (4) 24 (1991) 635–704. Crossref, Web of Science, Google Scholar