These lecture notes discuss notions of positivity for braids and links, their relation to complex plane curves, and how they provide strong restrictions on concordance properties of knots; e.g., sliceness. The notes are based on a three-hour mini-course delivered at Winter Braids XII.
Keywords: Braids, braid positive, quasipositive, strongly quasipositive, slice knots, slice-Bennequin-inequality, local Thom conjecture
Peter Feller  1
Peter Feller. Positive braids and concordance. Winter Braids Lecture Notes, Winter Braids XII, Tome 9 (2023), Exposé no. 2, 20 p.. doi: 10.5802/wbln.43
@article{WBLN_2023__9__A2_0,
author = {Peter Feller},
title = {Positive braids and concordance},
journal = {Winter Braids Lecture Notes},
note = {talk:2},
pages = {1--20},
year = {2023},
publisher = {Winter Braids School},
volume = {9},
doi = {10.5802/wbln.43},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/wbln.43/}
}
[Ale23] A Lemma on Systems of Knotted Curves, Proc. Nat. Acad. Sci. USA, Volume 9 (1923), pp. 93-95 | DOI | Zbl
[Art25] Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg, Volume 4 (1925) no. 1, pp. 47-72 | DOI | MR | Zbl
[BB05] Braids: a survey, Handbook of knot theory, Elsevier B. V., Amsterdam, 2005, pp. 19-103 | DOI | MR | Zbl
[Bir74] Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J., 1974, ix+228 pages (Annals of Mathematics Studies, No. 82) | MR | Zbl
[BK86] Plane algebraic curves, Birkhäuser Verlag, Basel, 1986, vi+721 pages (Translated from the German by John Stillwell) | MR | DOI | Zbl
[BO01] Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe, C. R. Acad. Sci. Paris Sér. I Math., Volume 332 (2001) no. 9, pp. 825-830 | DOI | MR | Zbl
[Don83] An application of gauge theory to four-dimensional topology, J. Differential Geom., Volume 18 (1983) no. 2, pp. 279-315 http://projecteuclid.org/euclid.jdg/1214437665 | MR | Zbl
[FM66] Singularities of -spheres in -space and cobordism of knots, Osaka Math. J., Volume 3 (1966), pp. 257-267 http://projecteuclid.org/euclid.ojm/1200691730 | MR | Zbl
[FQ90] Topology of 4-manifolds, Princeton Mathematical Series, 39, Princeton University Press, Princeton, NJ, 1990, viii+259 pages | MR | Zbl
[Fre82] The topology of four-dimensional manifolds, J. Differential Geom., Volume 17 (1982) no. 3, pp. 357-453 http://projecteuclid.org/... | MR | Zbl
[GS99] 4-manifolds and Kirby calculus, 20, American Mathematical Soc., 1999 | DOI | MR | Zbl
[KM93] Gauge theory for embedded surfaces. I, Topology, Volume 32 (1993) no. 4, pp. 773-826 | DOI | MR | Zbl
[KM94] The genus of embedded surfaces in the projective plane, Math. Res. Lett., Volume 1 (1994) no. 6, pp. 797-808 | MR | DOI | Zbl
[Mar36] Über die freie Äquivalenz geschlossener Zöpfe. Vortrag gehalten an der I. Internationalen Topologischen Konferenz den 5. 9. 1935, Rec. Math. Moscou, n. Ser., Volume 1 (1936), pp. 73-78 | Zbl
[Pap57] On Dehn’s lemma and the asphericity of knots, Proc. Nat. Acad. Sci. U.S.A., Volume 43 (1957), pp. 169-172 | DOI | MR | Zbl
[Ras10] Khovanov homology and the slice genus, Invent. Math., Volume 182 (2010) no. 2, pp. 419-447 (ArXiv:0402131v1 [math.GT]) | DOI | MR | Zbl
[Rud83a] Algebraic functions and closed braids, Topology, Volume 22 (1983) no. 2, pp. 191-202 | DOI | MR | Zbl
[Rud83b] Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv., Volume 58 (1983) no. 1, pp. 1-37 | DOI | MR | Zbl
[Rud83c] Constructions of quasipositive knots and links. I, Knots, braids and singularities (Plans-sur-Bex, 1982) (Monogr. Enseign. Math.), Volume 31, Enseignement Math., Geneva, 1983, pp. 233-245 | MR | Zbl
[Rud93] Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. (N.S.), Volume 29 (1993) no. 1, pp. 51-59 | DOI | MR | Zbl
[Rud99] Positive links are strongly quasipositive, Proceedings of the Kirbyfest (Berkeley, CA, 1998) (Geom. Topol. Monogr.), Volume 2, Geom. Topol. Publ., Coventry (1999), pp. 555-562 | DOI | MR
[Sma60] The generalized Poincaré conjecture in higher dimensions, Bull. Amer. Math. Soc., Volume 66 (1960), pp. 373-375 | DOI | MR | Zbl
[Sta62] The piecewise-linear structure of Euclidean space, Proc. Cambridge Philos. Soc., Volume 58 (1962), pp. 481-488 | MR | DOI | Zbl
[Tau87] Gauge theory on asymptotically periodic -manifolds, J. Differential Geom., Volume 25 (1987) no. 3, pp. 363-430 http://projecteuclid.org/euclid.jdg/1214440981 | MR | Zbl
[Vog90] Representation of links by braids: a new algorithm, Comment. Math. Helv., Volume 65 (1990) no. 1, pp. 104-113 | DOI | MR | Zbl
[Wal04] Singular points of plane curves, London Mathematical Society Student Texts, 63, Cambridge University Press, Cambridge, 2004, xii+370 pages | DOI | MR | Zbl
[Yam87] The minimal number of Seifert circles equals the braid index of a link, Invent. Math., Volume 89 (1987) no. 2, pp. 347-356 | DOI | MR | Zbl
Cité par Sources :

