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Links associated with generic immersions of graphs. (English) Zbl 1055.57006

An algebraic link is the link of a singularity of an algebraic curve, and a transverse \({\mathbb C}\)-link [L. Rudolph, Topology ’90, Contrib. Res. Semester Low Dimensional Topol., Columbus/OH (USA) 1990, Ohio State Univ. Math. Res. Inst. Publ. 1, 343-349 (1992; Zbl 0785.57002)] is a link which is represented as the transversal intersection of an algebraic curve and the unit 3-sphere in the 2-dimensional complex space. Though an algebraic link is a transverse \({\mathbb C}\)-link, the converse is not true [see L. Rudolph, Topology 22, 191–202 (1983; Zbl 0505.57003)]. A’Campo constructed links of divides in [N. A’Campo, Ann. Fac. Sci. Toulouse, VI. Sér., Math. 8, 5-23 (1999; Zbl 0962.32025)] as an extension of the class of algebraic links, where a divide is a generic relative immersion of a disjoint union of arcs and loops in a 2-disk, and showed in [N. A’Campo, Publ. Math., Inst. Hautes Étud. Sci. 88, 151-169 (1998; Zbl 0960.57007)] that any of the links of divides is ambient isotopic to a transverse \({\mathbb C}\)-link. In [L. Rudolph, Topology 22, 191-202 (1983; Zbl 0505.57003)], Rudolph also showed that a quasipositive link is a transverse \({\mathbb C}\)-link, where a quasipositive link is an oriented link which has a closed quasipositive braid diagram, that is a product of braids which are conjugates of positive braids, and the converse is also true by Boileau and Orevkov [M. Boileau and S. Orevkov, C. R. Acad. Sci., Paris, Sér. I, Math. 332, 825–830 (2001; Zbl 1020.32020)]. W. Gibson and M. Ishikawa [Topology Appl. 123, 609–636 (2002; Zbl 1028.57006)] constructed links of free divides, where a free divide is a generic non-relative immersion of a disjoint union of arcs in a 2-disk. In [T. Kawamura, Topology Appl. 125, 111–123 (2002; Zbl 1013.57003)], the author proved that links of divides and free divides are quasipositive and there exists quasipositive links which are not the links of any divide or free divide. In the paper under review, the author introduces the links of graph divides as an extension of the links of divides or free divides, where a graph divide is a generic non-relative immersion of a disjoint union of finite graphs and loops in a 2-disk, and proves that the links of graph divides are quasipositive. Furthermore the author proves that if a knot of a graph divide is a slice knot then it is a trivial knot. This implies that there exists a quasipositive link which is not a link of any graph divide. Actually the author gives an example of such a quasipositive link, the mirror image of the knot \(8_{20}\) in Rolfsen’s table which is quasipositive but a non-trivial slice knot.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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References:

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[2] N A’Campo, Generic immersions of curves, knots, monodromy and Gordian number, Inst. Hautes Études Sci. Publ. Math. (1998) · Zbl 0960.57007
[3] N A’Campo, Planar trees, slalom curves and hyperbolic knots, Inst. Hautes Études Sci. Publ. Math. (1998) · Zbl 0960.57008
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[5] W Gibson, Links and Gordian numbers associated with generic immersions of trees (2001) 27
[6] W Gibson, M Ishikawa, Links of oriented divides and fibrations in link exteriors, Osaka J. Math. 39 (2002) 681 · Zbl 1030.57007
[7] W Gibson, M Ishikawa, Links and Gordian numbers associated with generic immersions of intervals, Topology Appl. 123 (2002) 609 · Zbl 1028.57006
[8] M Hirasawa, Visualization of A’Campo’s fibered links and unknotting operation (2002) 287 · Zbl 1016.57006
[9] M Hirasawa, in preparation
[10] M Ishikawa, The \(\mathbbZ_2\)-coefficient Kauffman state model on divides, preprint
[11] T Kawamura, On unknotting numbers and four-dimensional clasp numbers of links, Proc. Amer. Math. Soc. 130 (2002) 243 · Zbl 0986.57006
[12] T Kawamura, Quasipositivity of links of divides and free divides, Topology Appl. 125 (2002) 111 · Zbl 1013.57003
[13] D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish (1990) · Zbl 0854.57002
[14] L Rudolph, Algebraic functions and closed braids, Topology 22 (1983) 191 · Zbl 0505.57003
[15] L Rudolph, Totally tangential links of intersection of complex plane curves with round spheres, Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 343 · Zbl 0785.57002
[16] L Rudolph, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. \((\)N.S.\()\) 29 (1993) 51 · Zbl 0789.57004
[17] T Shibuya, Some relations among various numerical invariants for links, Osaka J. Math. 11 (1974) 313 · Zbl 0291.55002
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