Visualization of A'Campo's fibered links and unknotting operation

Diagrams of divide links

“…An advantage of our approach is that we obtain a link diagram directly from a divide picture without deforming it into a so-called ordered Morse signed divide as in [8]. A similar method to draw diagrams was found by M. Hirasawa in [10]. Our diagrams are obviously symmetrical in a sense that the rotation of R 3 by 180…”

Links and gordian numbers associated with certain generic immersions of circles

“…Proposition 2.1 [Hirasawa 2002]. For any divide P, the diagram obtained by the above algorithm represents the link of P.…”

“…Let (respectively ) be the graph constructed by taking for each black (respectively white) region a vertex, and for each double point of P an edge connecting the vertices which are taken for black (respectively white) regions abutting at that point. The divide P is a slalom divide if either and is a tree (a slalom tree T P ) [A 'Campo 1998b;Hirasawa 2002]. A slalom divide is a connected divide such that every arc of P except near double points is accessible from the boundary of the unit disk D without passing through P [Ishikawa 2001b].…”

“…A divide P is a generic relative immersion in the unit disk (D, ∂ D) of a finite number of 1-manifolds, that is, copies of the unit interval (I, ∂ I ) and the unit circle [A 'Campo 1998a'Campo , 1999Hirasawa 2002;Ishikawa 2001b]. We also call the image of such an immersion a divide.…”

An infinite family of prime knots with a certain property for the clasp number

“…We call such a singular disk a clasp disk of We refer the reader to [4,6,7,11,12,13,14,18] for related topics of the clasp number. In this paper, we suppose that every link is in S 3 and oriented, and the notation of prime knots follows Rolfsen's book [16].…”