(1, 2) and weak (1, 3) homotopies on knot projections

Noboru Ito; Yusuke Takimura

doi:10.1142/s0218216513500855; 10.1142/s0218216514910015

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arXiv · 2004.06312

(1, 2) and weak (1, 3) homotopies on knot projections

Noboru Ito·Yusuke Takimura

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Abstract

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 1). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 3).

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Noboru Ito, Yusuke Takimura. 2020-04-14. (1, 2) and weak (1, 3) homotopies on knot projections. https://doi.org/10.1142/s0218216513500855%3B%2010.1142%2Fs0218216514910015

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