This talk is concerned with two joint works [1] and [2]. Every generic immersed spherical curve can be related to a simple closed curve by a finite sequence of the first, second, and third Reidemeister moves. We decompose the third Reidemeister move into just two kinds of moves: strong and weak third Reidemeister moves. In [1], we show that the strong (1, 3) homotopy equivalence class containing a simple closed curve is represented as any connected sum of finitely many spherical curves, each of which is either a simple closed curve, the curve that appears similar to ¡ç, or the trefoil projection. In [2], we show that the weak (1, 3) homotopy equivalence class containing a simple closed curve is represented as any connected sum of finitely many spherical curves, each of which is either a simple closed curve or the curve that appears similar to ¡ç. This talk will obtain the story from our starting point to the present and our motivation of this study.

Reference:
[1] N. Ito, Y. Takimura, and K. Taniyama, Strong and weak (1, 3) homotopies on knot projections, to appear in Osaka J. Math. [2] N. Ito and Y. Takimura, (1, 2) and weak (1, 3) homotopies on knot projections. J. Knot Theory Ramifications 22 (2013), 1350085 (14 pages).

This seminar will be given in Japanese and the presentation materials are shown in English. This seminar is also held as an CREST seminar.