Links associated with generic immersions of graphs

Abstract

As an extension of the class of algebraic links, A'Campo, Gibson, and Ishikawa constructed links associated to immersed arcs and trees in a two-dimensional disk. By extending their arguments, we construct links associated to immersed graphs in a disk, and show that such links are quasipositive.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-26.abs.htm

Links and gordian numbers associated with certain generic immersions of circles

“…Furthermore, by the results due to A 'Campo [1998a] and Gibson and Ishikawa [2002], the 4-dimensional clasp number and the gordian number of the link of an interval divide or a free interval divide are equal to the number of the double points, as the author remarked in [Kawamura 2002c[Kawamura , 2004.…”

Knot Theory of Complex Plane Curves

“…This construction has been extended-first by allowing non-proper edges ("free divides"; Gibson and Ishikawa, 2002a), more generally by allowing immersed unoriented circle components (Kawamura, 2002), and more generally yet (and most recently) by allowing immersions of graphs that are not 1-manifolds ("graph divides"; Kawamura, 2004).…”

“…If P is a free divide, possibly with circle components, then (5) L(P ) is strongly quasipositive (Kawamura, 2002). If P is a graph divide, then (6) L(P ) is quasipositive (Kawamura, 2004). …”

On the Thurston–Bennequin invariant of graph divide links

“…After that, W. Gibson defined tree divide links from the image of trees and determined their unknotting numbers, slice Euler characteristics, and Thurston-Bennequin invariant [6]. Finally, T. Kawamura introduced graph divide links, proved their quasipositivity and determined their slice Euler characteristics [10].…”

“…In this paper we will determine the maximal Thurston-Bennequin invariant for a class of links, called graph divide links. A graph divide, introduced by T. Kawamura in [10], is the image of a generic immersion ϕ: G → D of a disjoint sum G of intervals, circles and finite graphs into the unit disk D. We call the image of a vertex of G a vertex of ϕ(G) and a transversal crossing point produced by the immersion a double point of ϕ(G).…”