Links and Bifurcations in Nonsingular Morse–Smale Systems

Abstract

Wada's theorem classifies the set of periodic orbits in NMS systems on S3 as links, that can be written in terms of six operations. This characterization allows us to study the topological restrictions that links require to suffer a given codimension one bifurcation. Moreover, these results are reproduced in the case of NMS systems with rotational symmetries, introducing new geometrical tools.

Optical knots and contact geometry II. From Ranada dyons to transverse and cosmetic knots

“…Wada's results, summarized in Theorem 5.9., provide a formal description of the sequence of topological moves producing the iterated knots/links, beginning with the Hopf links. These moves are depicted in Fig.s 2-7 of the paper by Campos et al [61]. These are, still, just particular kinds of Kirby moves as explained in Appendix F. The description of these moves is totally disconnected from the description of dynamical bifurcations depicted in Fig.s 1-4 above.…”

“….s 2-7 of the paper by Campos et al[61]. These are, still, just particular kinds of Kirby moves as explained in Appendix F. The description of these moves is totally disconnected from the description of dynamical bifurcations depicted inFig.s 1-4above.…”

“…s 2-7 of paper by Campos et al[61]) it is clear that the moves suggested by Wada are just specific adaptations of Kirby (or Fenn-Rourke) moves.Appendix G Calculations of the various presentations for the fundamental group of the figure eight knot a) We begin with the set of relationsr 1 : bcb −1 = a, → c = b −1 ab (G.1)r 2 : ada −1 = b, → d = a −1 ba (G.2)r 3 : d −1 bd = c, → a −1 ba −1 b a −1 ba = c = b −1 ab (G.3) r 4 : c −1 ac = d → b −1 ab −1 a(b −1 ab) = d = a −1 ba (G.4) Noticing now that a −1 ba −1 = a −1 b −1 a and b −1 ab −1 = b −1 a −1 b we can rewrite eq. (G.3) as a −1 b −1 aba −1 ba = b −1 ab → a = ba −1 b −1 aba −1 bab −1 (G.5) and eq.…”