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Strong and weak (1, 3) homotopies on knot projections
by
Noboru Ito
Waseda Institute for Advanced Study
Coauthors: Yusuke Takimura and Kouki Taniyama
This talk introduces topics from two recent joint works [1] and [2] about knot projections (i.e., unoriented one component generic immersed spherical curves). As is well-known, every knot projection can be related to the trivial knot projection (i.e., the simple closed curve) by a finite sequence of the first, second, and third Reidemeister moves (RI, RII, and RIII). In 2008, Hagge and Yazinski showed the non-triviality of equivalence classes of knot projections under RI and RIII by giving the first example. However, even which knot projection can be trivialized by RI and RIII is still unknown. As a first step to consider this problem, we decomposed RIII into weak and strong RIII as the way of Viro. As a result, we determined the equivalence class containing the trivial knot projection under RI and weak RIII in [1] and that of RI and strong RIII in [2]. [1] (resp. [2]) determined other classes under RI and weak RIII (resp. strong RIII).
[1] N. Ito and Y. Takimura, (1, 2) and weak (1, 3) homotopies on knot projections, J. Knot Theory Ramifications 22 (2013), 1350085 (14 pages).
[2] N. Ito, Y. Takimura, and K. Taniyama, Strong and weak (1, 3) homotopies on knot projections, to appear in Osaka J. Math.
Date received: December 5, 2014
Copyright © 2014 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbjz-36.